Fundamental Formulas of Physics 1 (Menzel 0486605957)

FUNDAMENTAL FORMULAS

OF PHYSICS

EDITED BY

DONALD H. MENZEL DIRECTOR, HARVARD COLLEGE OBSERVATORY

In Two Volumes

VOLUME ONE

D O V E R P U B L I C A T I O N S , I N C .

N E W Y O R K

Copyright © 1960 by Dover Publications, Inc. All rights reserved under Pan American and Inter

national Copyright Conventions.

Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.

Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC 2.

This Dover edition, first published in 1960, is an unabridged and revised version of the work originally published in 1955 by Prentice-Hall, Inc. The first edition appeared in one volume, but this Dover edition is divided into two volumes.

International Standard Book Number: 0-486-60596-5

Library of Congress Catalog Card Number: 60-51149

Manufactured in the United States of America Dover Publications, Inc.

180 Varick Street New York, N. Y. 10014

P R E F A C E

A survey of physical scientists, made several years ago, indicated the need for a comprehensive reference book on the fundamental formulas of mathematical physics. Such a book, the survey showed, should be broad, covering, in addition to basic physics, certain cross-field disciplines where physics touches upon chemistry, astronomy, meteorology, biology, and electronics. . • *

The present volume represents an attempt to fill the indicated need. I am deeply indebted to the individual authors, who have contributed time and effort to select and assemble formulas within their special fields. Each author has had full freedom to organize his material in a form most suitable for the subject matter covered. In consequence, the styles and modes of presentation exhibit wide variety. Some authors considered a mere listing of the basic formulas as giving ample coverage. Others felt the necessity of adding appreciable explanatory text.

The independence of the authors has, inevitably, resulted in a certain amount of overlap. However, since conventional notation may vary for the different fields, the duplication of formulas should be helpful rather than confusing.

In the main, authors have emphasized the significant formulas, without attempting to develop them from basic principles. Apart from this omission, each chapter stands as a brief summary or short textbook of the field represented. In certain instances, the authors have included material not heretofore available.

The book, therefore, should fill needs other than its intended primary function of reference and guide for research. A student may find it a handy aid for review of familiar field or for gaining rapid insight into the techniques of new ones. The teacher will find it a useful guide in the broad field of physics. The chemist, the astronomer, the meteorologist, the biologist, and the engineer should derive valuable aid from the general sections as well as from the cross-field chapters in their specialties. For example, the chapter on Electromagnetic Theory has been designed to meet

v

vi PREFACE

the needs of both engineers and physicists. The handy conversion factors facilitate rapid conversion from Gaussian to MKS units or vice versa.

In a work of this magnitude, some errors will have inevitably crept in. I should appreciate it, if readers would call them to my attention.

DONALD H . MENZEL Harvard College Observatory

Cambridge, Mass.

CONTENTS

Chapter 1 : BASIC M A T H E M A T I C A L FORMULAS by Philip Franklin

1. ALGEBRA . . .

1.1. Quadratic equations . . . . 1 1.2. Logarithms 2 1.3. Binomial theorem . . / . 2 1.4. Multinomial theorem . . 2 1.5. Proportion . . . . . . . . . . . . 2

1.6. Progressions 3 1.7. Algebraic equations . . 3 1.8. Determinants 4 1.9. Linear equations . . . . . . 4

2 . TRIGONOMETRY

2.1. Angles . 2.2. Trigonometric functions 2.3. Functions of sums arid

differences 2.4. Addition theorems 2.5. Multiple angles . 2.6. Direction cosines . 2.7. Plane right triangle 2.8. Amplitude and phase . 2.9. Plane oblique triangle . 2.10. Spherical right -triangle 2.11. Spherical oblique

triangle

5 2.12. Hyperbolic functions . . 8 5 2.13. Functions of sums and

differences 9 5 2.14. Multiple arguments . . 9 5 2.15. Sine,cosine,and complex 6 exponential function 9 6 2.16. Trigonometric and hy-7 perbolic functions . . 9 7 2.17. Sine and cosine of com-7. plex arguments . . . . 10 8 2.18. Inverse functions and

logarithms 10

3. DIFFERENTIAL CALCULUS

3.1. The derivative 10 3.2. Higher derivatives . . . . 10 3.3. Partial derivatives . . . . 10 3.4. Derivatives of functions 11 3.5. Products 11 3.6. Powers and quotients . . 12 3.7. Logarithmic differentia

tion 12 3.8. Polynomials 12 3.9. Exponentials and loga

rithms 12 3.10. Trigonometric func

tions 12

3.11. Inverse trigonometric functions . . 12

3.12. Hyperbolic functions . . 13 3.13. Inverse hyperbolic func

tions 13 3.14. Differential 13 3.15. Total differential 13 3.16. Exact differential 13 3.17. Maximum and mini

mum values 14 3.18. Points of inflection . . . . 14 3.19. Increasing absolute

value 14

vii

viii CONTENTS

3.20. Arc length 14 3.21. Curvature 15 3.22. Acceleration in plane

motion 15 3.23. Theorem of the mean.. , 15

4. INTEGRAL CALCULUS

4.1. Indefinite integral . . . . 17 4.2. Indefinite integrals of

functions 18 4.3. Polynomials 18 4.4. Simple rational fractions 18 4.5. Rational functions . . . . 18 4.6. Trigonometric functions 19 4.7. Exponential and

hyperbolic functions.. 20 4.8. Radicals 20 4.9. Products 21 4.10. Trigonometric or expo

nential integrands . . 21 4.11. Algebraic integrands . . 22 4.12. Definite integral 22 4.13. Approximation rules . . 23 4.14. Linearity properties . . 24

5. DIFFERENTIAL EQUATIONS

5.1. Classification 28 5.2. Solutions . . 28 5.3. First order and first

degree 28 5.4. Variables separable . . . . 28 5.5. Linear in y 29 5.6. Reducible to linear . . . . 29 5.7. Homogeneous 30 5.8. Exact equations 30 5.9. First order and higher

degree : . . . 30 5.10. Equations solvable for p 30 5.11. Clairaut's form 30 5.12. Second order 31 5.13. Linear equations . . . . 32 5.14. Constant coefficients . . 33

6. VECTOR ANALYSIS

6.1. Scalars 39 6.2. Vectors 39 6.3. Components 39 6.4. Sums and products by

scalars 40

3.24. Indeterminate forms . . 15 3.25. Taylor's theorem . . . . 16 3.26. Differentiation of

integrals 17

1 7

4.15. Mean values 24 4.16. Inequalities 24 4.17. Improper integrals . . . . 25 4.18. Definite integrals of

functions 25 4.19. Plane area 25 4.20. Length of arc 25 4.21. Volumes 25 4.22. Curves and surfaces in

space 26 4.23. Change of variables in

multiple integrals . . 26 4.24. Mass and density . . . . 26 4.25. Moment and center of

gravity 27 4.26. Moment of inertia and

radius of gyration . . 27

2 8

5.15. Undetermined coefficients 34

5.16. Variation of parameters 35 5.17. The Cauchy - Euler

"homogeneous linear equation" 36

5.18. Simultaneous differential equations 36

5.19. First order partial differential equations.. 37

5.20. Second order partial differential equations . . 37

5.21. Runge-Kutta method of finding numerical solutions 39

, 3 9

6.5. The scalar or dot product, V,. • V2 40

6.6. The vector or cross product, V1X V2 40

6.7. The triple scalar product 41

CONTENTS ix

6.8. The derivative 41 6.9. The Frenet formulas.. 41 6.10. Curves with parameter t 42 6.11. Relative motion . . . . . . 42 6.12. The symbolic vector del 43 6.13. The divergence theorem 44

7. TENSORS

7.1. Tensors of the second rank 46

7.2. Summation convention.. 46 7.3. Transformation of

components 46 7.4. Matrix notation 46 7.5. Matrix products 46 7.6. Linear vector operation 47 7.7. Combined operators.... 47 7.8. Tensors from vectors.. 47 7.9. Dyadics 47 7.10. Conjugate tensor.

Symmetry 48

8. SPHERICAL HARMONICS

8.1. Zonal harmonics 51 8.2. Legendre polynomials . . 51 8.3. Rodrigues's formula . . 52 8.4. Particular values 52 8.5. Trigonometric polyno

mials 52 8.6. Generating functions . . 53 8.7. Recursion formula and

orthogonality 53 8.8. Laplace's integral 53

9. BESSEL FUNCTIONS

9.1. Cylindrical harmonics . . 55 9.2. Bessel functions of the

first kind '55 9.3. Bessel functions of the

second kind 56 9.4. Hankel functions 56 9.5. Bessel's differential

equation 57 9.6. Equation reducible to

Bessel's 57

6.14. Green's theorem in a plane 44

6.15. Stokes's theorem . . . . 44 6.16. Curvilinear coordinates 45 6.17. Cylindrical coordinates 45 6.18. Spherical coordinates.. 45 6.19. Parabolic coordinates.. 45

46

7.11. Unit, orthogonal, unitary 48

7.12. Principal axes of a symmetric tensor . . 49

7.13. Tensors in n-dimensions . . . . . . 49

7.14. Tensors of any rank . . 50 7.15. The fundamental tensor 50 7.16. ChristofTel three-index

symbols 50 7.17. Curvature tensor . . . . 51

5 1

8.9. Asymptotic expression" 53 8.10. Tesseral harmonics.... 53 8.1 L Legendre's associated

functions 54 8.12. Particular values 54 8.13. Recursion formulas . . 54 8.14. Asymptotic expression 54 8.15. Addition theorem 55 8.16. Orthogonality 55

5 5

9.7. Asymptotic expressions. 58 9.8. Order half an odd integer 58 9.9. Integral representation. . 59 9.10. Recursion formula . . . . 59 9.11. Derivatives 59 9.12. Generating function . . 59 9.13. Indefinite integrals . . 59 9.14. Modified Bessel

functions •. 59

X CONTENTS

10 . T H E HYPERGEOMETRIC FUNCTION

10.1. The hypergeometric equation 60

10.2. The hypergeometric series 60

10.3. Contiguous functions . . 61 10.4. Elementary functions . . 62 10.5. Other functions 62 10.6. Special relations 62

11 . LAGUERRE FUNCTIONS

11.1. Laguerre polynomials. . 64 11.2. Generating function . . 64 11.3. Recursion formula . ; . . 64 11.4. Laguerre functions . . 65

12 . HERMITE FUNCTIONS

12.1. Hermite polynomials . . 65 12.2. Generating function . . 66

13. MISCELLANEOUS FUNCTIONS . .

13.1. The gamma function . . 66 13.2. Functional equations . . 67 13.3. Special values 67 13.4. Logarithmic derivative 67 13.5. Asymptotic expressions 67 13.6. Stirling's formula . . . . 68

14. SERIES

14.1. Bernoulli numbers . . . . 69 14.2. Positive powers . . . . . . 70 14.3. Negative powers . . . . . 70 14.4. Euler - Maclaurin sum

formula 70 14.5. Power series 70 14.6. Elementary functions. . 71 14.7. Integrals 72 14.8. Expansions in rational

fractions 72 14.9. Infinite products for the

sine and cosine . . . . 73 14.10. Fourier's theorem for

periodic functions . 73

6 0

10.7. Jacobi polynomials or hypergeometric polynomials . . 63

10.8. Generalized hypergeometric functions . . . . 63

10.9. The confluent hypergeometric function . . 64

6 4

11.5. Associated Laguerre polynomials 65

11.6. Generating function . . 65 11.7. Associated Laguerre

functions 65

6 5

12.3. Recursion formula . . . . 66 12.4. Hermite functions . . . . 66

6 6

13.7. The beta function 68 13.8. Integrals 68 13.9. The error integral 69 13.10. The Riemann zeta

function 69

6 9

14.11. Fourier series on an interval 74

14.12. Half-range Fourier series 74

14.13. Particular Fourier series 74

14.14. Complex Fourier series 76 14.15. The Fourier integral

theorem 76 14.16. Fourier transforms . . 77 14.17. Laplace transforms . . 77 14.18. Poisson's formula . . . . 77 14.19. Orthogonal functions. 78 14.20. Weight functions 78

CONTENTS xi

15 . ASYMPTOTIC EXPANSIONS 7 9

15.1. Asymptotic expansion. 79 15.2. Borel's expansion . . . . 79

15.3. Steepest descent. 80

16 . LEAST SQUARES 8 0

16.1. Principle of least squares 80 16.2. Weights 81 16.3. Direct observations . . 81

16.4. Linear equations . . . . 81 16.5. Curve fitting 81 16.6. Nonlinear equations . . 82

17. STATISTICS 8 2

17.1. Average . 82 17.7. Mean absolute error . . 84 17.2. Median . 82 17.8. Probable error 84 17.3. Derived averages . . . . 83 17.9. Measure of dispersion. 84 17.4. Deviations . 8? 17.10. Poisson's distribution. 84 17.5. Normal law / 83 17.11. Correlation coefficient 85 17.6. Standard deviation . . 83

18. MATRICES 8 5

18.1. Matrix 85 18.2. Addition 86 18.3. Multiplication 86 18.4. Linear transformations 86 18.5. Transposed matrix . . . . 87 18.6. Inverse matrix 87

18.7. Symmetry 87 18.8. Linear equations . . . . 87 18.9. Rank 88 18.10. Diagonalization of ma

trices 88

19. GROUP THEORY

19.1. Group 89 19.2. Quotients 90 19.3. Order 90 19.4. Abelian group 90 19.5. Isomorphy 90

19.6. Subgroup 90 19.7. Normal divisor 90 19.8. Representation 91 19.9. Three-dimensional ro

tation group 91

8 9

2 0 . ANALYTIC FUNCTIONS , 9 3

20.1. Definitions 93 20.2. Properties 94 20.3. Integrals 94 20.4. Laurent expansion . . . . 94

20.5. Laurent expansion about infinity 95

20.6. Residues 95

2 1 . INTEGRAL EQUATIONS 9 5

21.1. Fredholm integral equations . . 96

21.2. Symmetric kernel 97 21.3. Volterra integral equa

tions 98 21.4. The Abel integral equa

tion 99

21.5. Green's function 99 21.6. The Sturm - Liouville

differential equations 100 21.7. Examples of Green's

function 102

xii CONTENTS

Chapter 2 : S T A T I S T I C S 1 0 7

by Joseph M. Cameron

1. INTRODUCTION 1 0 7

1.1. Characteristics of a meas- 1.2. Statistical estimation . . 108 urement process . . . . 107 1.3. Notation 108

2 . STANDARD DISTRIBUTIONS 1 0 9

2.1. The normal distribution 109 2.5. The x2 distribution 110

2.2. Additive property . . . . 109 2.6. Student's t distribution. I l l 2.3. The logarithmic-normal 2.7. The F distribution 111

distribution 110 2.8. Binomial distribution . . I l l 2.4. Rectangular distribution 110 2.9. Poisson distribution . . . . 112

3. ESTIMATORS OF THE LIMITING MEAN 1 1 2

3.1. The average or arithmetic 3.2. The weighted average . . 113 mean 112 3.3. The median 113

4 . MEASURES OF DISPERSION 1 1 3

4.1. The standard deviation . 1 1 3 4.3. Average deviation,

4.2. The variance s2 = _L I # • £ I 115 n 1=1

n 2 Xi — x)2/(n — 1) . . . 115 4.4. Range : difference be-

t = c l tween largest and smallest value in a set of observations, x, .==#, . , . . 116

' («) (i)

5. T H E FITTING OF STRAIGHT LINES 1 1 6

5.1. Introduction 116 5.2. The case of the underlying physical law . . . . 116

6 . LINEAR REGRESSION 1 2 0

6.1. Linear regression 120

7. T H E FITTING OF POLYNOMIALS 121

7.1. Unequal intervals be- 7.3. Fitting the coefficients of tween the x's 121 a function of several

7.2. The case of equal inter- variables 124 vals between the x's— 7.4. Multiple regression . . . . 124 the method of orthogonal polynomials . . 123

CONTENTS Xlll

1 2 5 8. ENUMERATIVE STATISTICS

8.1. Estimator of parameter of binomial distribution 125

8.2. Estimator of parameter of

9. INTERVAL ESTIMATION

9.1. Confidence interval for parameters 126

9.2. Confidence interval for the mean of a normal distribution . 127

9.3. Confidence interval for the standard deviation of a normal distribu-

1 0 . STATISTICAL TESTS OF HYPOTHESIS

10.1. Introduction . 129 10.2. Test of whether the

mean of a normal distribution is greater than a specified value 129

10.3. Test of whether the mean of a normal distribution is different from some specified value 130

10.4. Test of whether the mean of one normal distribution is greater than the mean of another normal distribution 130

1 1 . ANALYSIS OF VARIANCE

11.1. Analysis of variance . . 134

12 . DESIGN OF EXPERIMENTS

12.1. Design of experiments. 137

13. PRECISION AND ACCURACY

13.1. Introduction 137 13.2. Measure of precision . . 137

14. L A W OF PROPAGATION OF ERROR .

14.1. Introduction 138 14.2. Standard deviation of a

ratio of normally distributed variables... 140

Poisson distribution . . 125 8.3. Rank correlation coeffi

cient 126

1 2 6

tion 127 9.4. Confidence interval for

slope of straight line 128 9.5. Confidence interval for

intercept of a straight line 128

9.6. Tolerance limits 128

1 2 9

10.5. Test of whether the means of two normal distributions differ . 131

10.6. Tests concerning the parameters of a linear law 131

10.7. Test of the homogeneity of a set of variances 132

10.8. Test of homogeneity of a set of averages . . . . 132

10.9. Test of whether a correlation coefficient is different from zero . 133

10.10. Test of whether the • correlation coeffi

cient p is equal to a specified value 133

1 3 4

1 3 7

1 3 7

13.3. Measurement of accuracy 138

1 3 8

14.3. Standard deviation of a product of normally distributed variables 140

XIV CONTENTS

Chapter 3 : N O M O G R A M S 141 by Donald H. Menzel

1. NOMOGRAPHIC SOLUTIONS . 141

1.1. A nomogram (or nomograph) 141

Chapter 4 : PHYSICAL CONSTANTS 1 4 5 by Jesse W. M. DuMond and E. Richard Cohen

1. CONSTANTS AND CONVERSION FACTORS T OF ATOMIC AND NUCLEAR PHYSICS 1 4 5

2 . TABLE OF LEAST-SQUARES-ADJUSTED OUTPUT VALUES 148

Chapter 5 : CLASSICAL MECHANICS 1 5 5 by Henry Zatzkis

1. MECHANICS OF A SINGLE MASS POINT AND A SYSTEM OF MASS POINTS 1 5 5

1.1. Newton's laws of motion and fundamental motions 155

1.2. Special cases 157 1.3. Conservation laws 159 1.4. Lagrange equations of

the second kind for arbitrary curvilinear coordinates 160

1.5. The canonical equations of motion 161

1.6. Poisson brackets 162 1.7. Variational principles . . 163

1.8. Canonical transformations . . 164

1.9. Infinitesimal contact transformations 167

1.10. Cyclic variables 168 1.11. Transition to wave me

chanics. The optical-mechanical analogy. 171

1.12. The Lagrangian and Hamiltonian formalism for continuous systems and fields . . 174

Chapter 6 : SPECIAL THEORY OF RELATIVITY . , 178 by Henry Zatzkis

1. T H E KINEMATICS OF THE SPACE-TIME CONTINUUM 178

1.1. The Minkowski "world" 178 1.2. The Lorentz transfor

mation 179

1.3. Kinematic consequences of the Lorentz transformation 181

2 . DYNAMICS 1 8 6

2.1. Conservation laws . . . . 186 2.3. Relativistic force 189 2.2. Dynamics of a free mass 2.4. Relativistic dynamics . . 189

point 188 2.5. Gauge invariance 191

CONTENTS XV

2.6. Transformation laws for the field strengths . . .

2.7. Electrodynamics in moving, isotropic ponderable media (Minkowski's equation)

2.8. Field of a uniformly moving point charge

192

192 2.9.

in empty space. Force between point charges moving with the same constant velocity . . . . 194

The stress energy tensor and its relation to the conservation laws . . . . 196

3. MISCELLANEOUS APPLICATIONS '. 1 9 7

3.1. Theponderomotiveequation . 197

3.2. Application to electron optics 198

3.3. Sommerfeld's theory of the hydrogen fine structure 200

4 . SPINOR CALCULUS 2 0 1

4.1. Algebraic properties . . . 201 4.2. Connection between spi-

nors and world tensors 203

4.3. Transformation laws for mixed spinors of second rank. Relation

between spinor and Lorentz transformations 204

4.4. Dual tensors 206 4.5. Electrodynamics of emp

ty space in spinor form 208

5. FUNDAMENTAL RELATIVISTIC INVARIANTS 2 0 9

Chapter 7 : T H E G E N E R A L T H E O R Y O F R E L A T I V I T Y . . . . . . . 2 1 0 liy Henry Zatzkts

1. MATHEMATICAL BASIS OF GENERAL RELATIVITY 2 1 0

1.1. Mathematical introduction 210

1.2. The field equations 211 1.3. The variational principle 212 1.4. The ponderomotive law 213

1.5. The Schwarzschild solution 214

1.6. The three "Einstein effects" 214

Chapter 8 : H Y D R O D Y N A M I C S A N D AERODYNAMICS 2 1 8 by Max M. Munk

1. ASSUMPTIONS AND DEFINITIONS 2 1 8

2 . HYDROSTATICS 2 1 9

3. KINEMATICS 2 2 0

4 . THERMODYNAMICS 2 2 3

5. FORCES AND STRESSES 2 2 3

xvi CONTENTS

6 . DYNAMIC EQUATIONS 2 2 4

7. EQUATIONS OF CONTINUITY FOR STEADY POTENTIAL F L O W OF

NONVISCOUS FLUIDS 2 2 5

8. PARTICULAR SOLUTIONS OF LAPLACE'S EQUATION 2 2 7

9 . APPARENT ADDITIONAL MASS 2 2 8

10 . AIRSHIP THEORY . . 2 3 0

1 1 . W I N G PROFILE CONTOURS; TWO-DIMENSIONAL F L O W WITH CIRCU

LATION 2 3 1

1 2 . AIRFOILS IN THREE DIMENSIONS 2 3 2

13 . THEORY OF A UNIFORMLY LOADED PROPELLER DISK 2 3 3

14 . FREE SURFACES 2 3 3

15 . VORTEX M O T I O N . 2 3 4

16 . WAVES 2 3 5

17. MODEL RULES . 2 3 7

18 . VISCOSITY 2 3 8

19 . GAS FLOW, O N E - AND TWO-DIMENSIONAL 2 3 9

2 0 . GAS FLOWS, THREE DIMENSIONAL 2 3 9

2 1 . HYPOTHETICAL GASES 2 3 9

2 2 . SHOCKWAVES 2 4 0

2 3 . COOLING . 2 4 1

2 4 . BOUNDARY LAYERS 2 4 2

Chapter 9 : B O U N D A R Y V A L U E P R O B L E M S I N M A T H E M A T I C A L P H Y S I C S 2 4 4

by Henry Zatzkis

1. T H E SIGNIFICANCE OF THE BOUNDARY 2 4 4

1.1. Introductory remarks . . 244 1.7. The one - dimensional 1.2. The Laplace equation . . 244 wave equation 253 1.3. Method of separation of 1.8. The general eigenvalue

variables 246 problem and the high-1.4. Method of integral er-dimensional wave

equations 249 equation 255 1.5. Method of Green's func- 1.9. Heat conduction equa

tion 249 tion 261 1.6. Additional remarks about 1.10. Inhomogeneous differ

tile two - dimensional ential equations 263 area. 251

CONTENTS

Chapter 1 0 : H E A T A N D T H E R M O D Y N A M I C S . . . . . . . . . . . . . 2 6 4 by Percy W. Bridgman

1. FORMULAS OF THERMODYNAMICS 2 6 4

1.1. Introduction 264 1.2. The laws of thermody

namics 265 1.3. The variables 266 1.4. One-component systems 266 1.5. One-component, usually

two-phase systems . . . 268 1.6. Transitions of higher

orders 269 1.7. Equations of state 270 1.8. One - component, two-

variable systems, with

dW = Xdy 271 1.9. One - component, two-

variable systems, with dW = Xdy 271

1.10. One-component, multi-variable systems, with dW=X Xidyi..... Ill

1.11. Multicomponent, multivariate systems, dW = pdv 271

1.12. Homogeneous systems 273 1.13. Heterogeneous systems 275

Chapter 11 : STATISTICAL MECHANICS 2 7 7 by Donald H. Menzel

1. STATISTICS OF MOLECULAR ASSEMBLIES 2 7 7

1.1. Partition functions 277 1.2. Equations of state 280 1.3. Energies and specific

heats of a one-component assembly 281

1.4. Adiabatic processes . . . . 281 1.5. Maxwell's and Boltz-

mann's laws 282 1.6. Compound and disso

ciating assemblies . . . . 282 1.7. Vapor pressure 283

1.8.

1.9.

Convergence of partition functions 284

Fermi-Dirac and Bose-Einstein statistics . . . . 284

1.10. Relativistic degeneracy 286 1.11. Dissociation law for

new statistics 287 1.12. Pressure of a degenerate

gas 288 1.13. Statistics of light quanta 289

Chapter 1 2 : KINETIC THEORY OF GASES : VISCOSITY, THERMAL CONDUCTION, AND DIFFUSION. . . 2 9 0

by Sydney Chapman

1. PRELIMINARY DEFINITIONS AND EQUATIONS FOR A MIXED GAS, N O T IN EQUILIBRIUM 2 9 0

1.1. Definition of r, x, y, z> t, 1.5. The velocity distribution dr 290 function/ 291

1.2. Definition of m, w, />, w1 0, 1.6. Mean values of velocity ni2, m0 290 functions 291

1.3. The external forces Fy 1.7. The mean mass velocity Xy Y, Z, «A 291 c0) u0y v0> w0. 291

1.4. Definition of c, u, vt zv, 1.8. The random velocity C; dc 291 U, V, W\ dC 292

xviii CONTENTS

1.9. The "heat" energy of a molecule E: its mean value E 292

1.10. The molecular weight W and the constants ™u, JVr 2 9 2

1.11. The constants / , ky R . . 292 1.12. The kinetic theory tem

perature T 293 1.13. The symbols cvy Cv... 293

1.14. The stress distribution, 293

1.15. The hydrostatic pressure p ; the partial pressures p l f p 2 294

1.16. Boltzmann's equation f o r / . . 294

1.17. Summational invariants 294 1.18. Boltzmann's H theorem 294

2 . RESULTS FOR A GAS IN EQUILIBRIUM 2 9 5

2.1. Maxwell's steady-state solutions. . . 295

2.2. Mean values when / is Maxwellian 296

2.3. The equation of state for a perfect gas 296

2.4. Specific heats 297

2.5. Equation of state for an imperfect gas . . 297

2.6. The free path, collision frequency, collision interval, and collision energy (perfect gas). . 298

3. NONUNIFORM GAS 2 9 9

3.1. The second approxima- 3.3. Diffusion 300 t ionto / 299 3.4. Thermal conduction 301

3.2. The stress distribution 300

4 . T H E GAS COEFFICIENTS FOR PARTICULAR MOLECULAR M O D E L S . . 3 0 1

4.1. Models a to d 301 4.3. The first approximation 4.2. Viscosity and thermal to D12 303

conductivity A for a 4.4. Thermal diffusion 304 simple gas 302

5. ELECTRICAL CONDUCTIVITY IN A NEUTRAL IONIZED GAS WITH OR WITHOUT A MAGNETIC FIELD 3 0 4

5.1. Definitions and symbols 304 5.3. Slightly ionized gas . . . 305 5.2. The diameters as 305 5.4. Strongly ionized gas . . 306

Chapter 1 3 : E L E C T R O M A G N E T I C THEORY 3 0 7 by Nathaniel H. Frank and William Tobocman

1. DEFINITIONS AND FUNDAMENTAL LAWS 3 0 7

1.1. Primary definitions . . . . 307 1.6. Vector and scalar poten-1.2. Conductors 308 tials 310 1.3. Ferromagnetic materials 309 1.7. Bound charge 310 1.4. Fundamental laws 309 1.8. Amperian currents 311 1.5. Boundary conditions.. 309

CONTENTS xix

2 . ELECTROSTATICS

2.1. Fundamental l a w s . . . . . 311 2.2. Fields of some simple

charge distributions.. 312 2.3. Electric multipoles; the

double layer . . . . . . . . 312 2.4. Electrostatic boundary

value problems 314 2.5. Solutions of simple elec-

3. MAGNETOSTATICS

3.1. Fundamental laws 320 3.2. Fields of some simple

current distributions 321 3.3. Scalar potential for mag

netostatics; the magnetic dipole ! . 322

4 . ELECTRIC CIRCUITS

4.1. The quasi - stationary approximation 328

4.2. Voltage and impedance 328 4.3. Resistors and capacitors

in series and parallel

5. ELECTROMAGNETIC RADIATION

5.1. Poynting's theorem . . . . 330 5.2. Electromagnetic stress

and momentum 331 5.3. The Hertz vector; electro

magnetic waves 332 5.4. Plane waves 335 5.5. Cylindrical waves 336 5.6. Spherical waves 337 5.7. Radiation of electromag

netic waves; the oscillating dipole 339

3 1 1

trostatic boundary value problems 314

2.6. The method of images 315 2.7. Capacitors 318 2.8. Normal stress on a con

ductor. . . ./ 319 2.9. Energy density of the

electrostatic field . . . . 319

3 2 0

3.4. Magnetic multiples . . . . 323 3.5. Magnetostatic boundary-

value problems 3251 3.6. Inductance 3261 3.7: Magnetostatic energy

density 327

3 2 8

connection 328 4.4. KirchhofFs rules 328 4.5. Alternating current

circuits 329

3 3 0

5.8. Huygen's principle . . . . 340 5.9. Electromagnetic waves

at boundaries in dielectric media 341

5.10. Propagation of electromagnetic radiation in wave guides 342

5.11. The retarded potentials; the Lienard-Wiechert potentials; the self-force of electric charge 344

Chapter 1 4 : ELECTRONICS 3 5 0 by Emory L. Chaffee

1. ELECTRON BALLISTICS 3 5 0

1.1. Current 350 1.3. Energy of electron.... 351 1.2. Forces on electrons . . . . 350 1.4. Electron orbit 351

2 . SPACE CHARGE 3 5 2

2.1. Infinite parallel planes . . 352 2.2. Cylindrical electrodes.. 352

XX CONTENTS

3. EMISSION OF ELECTRODES 353

3.1. Thermionic emission. . . 353 3.2. Photoelectric emission . . 353

4. FLUCTUATION EFFECTS 354

4.1. Thermal noise 354 4.2. Shot noise 354

Chapter 15 : S O U N D A N D A C O U S T I C S 355

by Philip M. Morse

1. SOUND AND ACOUSTICS 355

1.1. Wave equation, definition 355 1.2. Energy, intensity 356 1.3. Plane wave of sound . . . . 356 1.4. Acoustical constants for

various media 357 1.5. Vibrations of sound pro

ducers; simple oscillator 357

1.6. Flexible string under tension 357

1.7. Circular membrane under tension 358

1.8. Reflection of plane sound waves, acoustical impedance 359

1.9. Sound transmission through ducts 359

1.10. Transmission through long horn 360

1.11. Acoustical circuits . . . . 360 1.12. Radiation of sound from

a vibrating cylinder 361 1.13. Radiation from a simple

source 361 1.14. Radiation from a dipole

source 362 1.15. Radiation from a piston

in a wall 362 1.16. Scattering of sound

from a cylinder . . . . 362 1.17. Scattering of sound

from a sphere 363 1.18. Room acoustics 363

I N D E X

F U N D A M E N T A L

F O R M U L A S

of

P H Y S I C S

Chapter 1

BASIC MATHEMATICAL FORMULAS B Y P H I L I P F R A N K L I N

Professor of Mathematics

Massachusetts Institute of Technology

Certain parts of pure mathernatics are easily recognized as necessary tools of theoretical physics, and the results most frequently applied may be summarized as a compilation of formulas and theorems. But any physicist who hopes or expects that any moderate sized compilation of mathematical results will satisfy all his needs is doomed to disappointment. In fact no collection, short of a library, could begin to fulfill all the demands that the physicist will make upon the pure mathematician. And these demands grow constantly with the increasing size of the field of mathematical physics.

The following compilation of formulas is, therefore, intended to be representative rather than comprehensive. T o conserve space, certain elementary formulas and extensive tables of indefinite integrals have been omitted, since these are available in many well-known mathematical handbooks.

A select list of reference books, arranged by subject matter, is given at the end of Chapter 1. This bibliography may assist the reader in checking formulas, or in pursuing details or extensions beyond the material given here.

As a preliminary check on the formulas to be included, a tentative list was submitted to a number of physicists. These included all the authors of the various chapters of this book. They contributed numerous suggestions, which have been followed as far as space requirements have permitted.

are

1. Algebra

1 . 1 . Quadratic equations. If a ^ 0 , the roots of

ax2 -f- bx + c = 0

_ -b + (b2 - Aac)1^ _ -2c

(1)

( 2 ) 2a b -+(b2 -Aacfl2

-b-(b2 -Aacyl* 2c (3) 2a -b + (b2 - Aacf'2

1

2 BASIC MATHEMATICAL FORMULAS § 1.2

When b2 > | 4ac | , the second form for x1 when 6 > 0, and for x2 when b < 0 is easier to compute with precision, since it involves the sum, instead of the difference, of two nearly equal terms.

1.2* Logarithms. Let In represent natural logarithm, or loge where e = 2.71828. Then for M > 0, N > 0 :

In MN = In M + In iV, In Mv = p In M, In 1 = 0 |

In = l n M - l n i V , l n - ^ - = - l n i V , In « = 1

In this book log 1 0 will be written simply log. For conversion between base e and base 10

In M= 2.3026 log AT, log M = 0.43429 In ikf (2)

1 .3 . Binomial theorem. For n any positive integer,

(fl + by = an + na^b + n^n ~ ^ ^ - 2 6 2

+ ttfo-^H*-2) ^n-3£3 + ... + + 6* (1) or

r=0 r=0'm\n r

where n\ (factorial n) is defined by

» ! = 1-2-3- ... - (N —1)« for n>l and 1! = 1, 0! = 1 (3)

1.4. Multinomial theorem. For n any positive integer, the general term in the expansion of (a± + a2 + ... +<z&)w i s

. , n ! r fl/W8 • • • a k k (1)

w h e r e T u r 2 , ... rf c are positive integers such that ^ + r 2 + ... + rf c = rc.

1 .5 . Proportion

~ = or ~ = — ^ a n 0 H

If a/4 = £/J3 = £/C = k, then for any weighting factors p, q, r each fraction equals

_ pa + qb + rc pA + qB + rC K L )

§ 1.6 BASIC MATHEMATICAL FORMULAS 3

1.6. Progressions* Let / be the last term. Then the sum of the arithmetic progression to n terms

s = a + (a + d) + (a + 2d)+ ... + \a + (n- \)d]

is

s = ^(a + l), (1)

where / = a + (n — l)d, and the sum of the geometric progression to n terms

s = a + ar + ar2 + ... - f arn~x = ^ ^ ~7) ^

1.7. Algebraic equations. The general equation of the nth degree

P(x) = a0xn - f a^-1 + a2xn~2 + ... + an_±x + an = 0 (1)

has n roots. If the roots of P(x) = 0, or zeros of P(x), are r 1 ? r 2, r n , then

P(«) = a0(x — (a? — r2) ...(x — rn)

and the symmetric functions of the roots

v 2 - . . ^ = ( - i r ^ (2)

If m roots are equal to r, then r is a multiple root of order m. In this case P(x) = (x — r)mQ1(x) and P'(«) = (x - r ) ™ - 1 ^ * ) , where P'(x) = dP/dx, § 3.1. Thus r will be a multiple root of P\x) = 0 of order ra — 1. All multiple or repeated roots will be zeros of the greatest common divisor of P(x) and P'(x).

If r is known to be a zero of P(x)y so that P(r) = 0, then (# — r) is a factor of P(f), and dividing P(#) by (x — r) will lead to a depressed equation with degree lowered by one.

When P(a) and P(b) have opposite signs, a real root lies between a and 6. The interpolated value c — a — P(«) (Z> — a)j[P(b) — P(fl)], and calculation of P(c) will lead to new closer limits.

If cr is an approximate root, c2 = cx —Pc)jP'c) is Newton's improved approximation. We may repeat this process. Newton's procedure can be applied to transcendental equations. If an approximate complex root can be found, it can be improved by Newton's method.

4 BASIC MATHEMATICAL FORMULAS §1 .8

1.8. Determinants. The determinant of the nth order

D: 21 a, 22

unl an2

is defined to be the sum of n\ terms

a)

D = S (±) aua2ia3k ... anl (2)

In each term the second subscripts ijk...l are one ortder or permutation of the numbers 123...n. The even permutations, which contain an even number of inversions, are given the plus sign. The odd permutations, which contain an odd number of inversions, are given the minus sign.

The cofactor Au of the element aj is ( — t i m e s the determinant of the (n — l)st order obtained from D by deleting the zth row and7th column.

The value of a determinant D is unchanged if the corresponding rows and columns are interchanged, or if, to each element of any row (or column), is added m times the corresponding element in another row (or column).

If any two rows (or columns) are interchanged, D is multiplied by (— 1). If each element of any one row (or column) is multiplied by m, then D is multiplied by m.

If any two rows (or columns) are equal, or proportional, D = 0.

D = axiAxi + a2jA2j + ... + ^njAni

0 — aÂxh + a2jA2Jc + ... + anjAniv k^j

where j and k are any two of the integers 1, 2, n.

(3)

(4)

1.9. Linear equations. The solution of the system

&llxl 4 ~ &12X2 ~\~ "• ~ k amxn — ci

a21Xl H~ 22 2 H " ••• - 4 " a2nXn ~ C2

anlXl + an2X2 + • • • + annXn — Cn

is unique if the determinant of §1.8, D ^ 0. The solution is

D ' D ' D

(1)

(2)

where Ck is the nth order determinant obtained from D by replacing the elements of its kth column a11c, a21c> anlc by cv c2, cn.

§2.1 BASIC MATHEMATICAL FORMULAS 5

When all the ci = 0, the system is homogeneous. In this case, if D = 0, but not all the Ai are zero, the ratios of the xt satisfy

x l 2 xn /o\ A ~ A ~ "' ~ A ^ ' ^12

A homogeneous system has no nonzero solutions if D 7^ 0.

2. Trigonometry

2*1. Angles. Angles are measured either in degrees or in radians, units such that

2 right angles ==180 degrees = IT radians where TT = 3.14159. For conversion between degrees and radians,

1 degree = 0.017453 radian; 1 radian = 57.296 degrees

2.2. Trigonometric functions. For a single angle A, the equations

A sin A . cos A \ tan A = j , cot A = - — j , cos A sin A f m

1 1 I * ' sec A = -=-, esc A = ——T cos A sin A J

define the tangent, cotangent, secant, and cosecant in terms of the sine and cosine. Between the six functions we have the relations

sin2 A + cos 2 A = \, cot A = —^— ) tan A > [A)

sec2 A = l+ tan2 A, esc2 A = \+ cot 2 A )

2.3. Functions of sums and differences

sin (A + B) = sin A cos B + cos A sin B sin A — B) = sin A cos B — cos A sin B cos (A + B) = cos -4 cos 5 — sin 4 sin 5 cos ( 4 — B) = cos 4 cos B sin A sin -B

(I)

, . , tan 4 + tan B , . n . tan A — tan B _x tan ( 4 + 5 ) = , tan (i4 — B) = — -= ^ (2) v ^ ; 1 — tan A tan v ' 1 + tan A tan B v 7

2.4. Addition theorems

sin A + sin B = 2 sin \(A + £ ) cos|(^4 — J5) sin 4 — sin B == 2 cos ^ ( 4 + £ ) sin %(A —B) cos 4 + cos 5 = 2 cos \ (A + £ ) cos \ (A — B) cos A — cos £ = —2 sin ^ ( 4 + 5 ) sin ^ ( 4 — B)

(1)


2.5. Multiple angles

sin 2A '= 2 sin A cos A, cos 2A = cos2 A — sin2 A (1)

sin2 A = \ — J cos 2 4 , • cos 2 4 = . | + -| cos 2A (2)

. 4 / l - c o s 4 A , / l + cos 4.-S M T = ± V — - c o s T = ± y j (3)

tan 2 4 2 tan 4

1 - tan2 J '

A • 1 —- cos 4 1 — cos A sin 4 tan — = ± ^ j + c o s A = s i n ^ = l + cos ^

sin 3 4 = 3 sin 4 — 4 sin3 A, cos 3 4 = 4 cos 3 4 — 3 cos A

sin 4 4 = sin 4(8 cos 3 A — 4 cos 4 ) , c o s 4 4 = 8 cos 4 4 — 8 cos 2 4 + 1

(4)

(5)

(6)

(7) sin4 A = | ( 3 — 4 cos 2A + cos 44) ,

cos 4 A = 1 ( 3 + 4 cos 2A + cos 4A)

cos nA cos 7ra4 = ^ cos (« — m)4 + \ cos (w + m)A (8)

sin ; /4 sin *nA = \ cos (w — m)A — \ cos (n + #z)4 (9)

cos nA sin #z4 = -| sin (n + #z)4 — \ sin (« — #z)4 (10)

2.6. Direction cosines. The direction cosines of a line in space are the cosines of the angles a, /?, y which a parallel line through the origin O makes with the coordinate axes OX, O Y, OZ. For the line segment P^P2, j ° m m £ the points P 2 = (^y^Zj) and P2 = (x2iy2lz2) the direction cosines are

c o s a = x z ~ x \ C 0 S ^ = Z i _ = A ( C 0 S r = = f i J ^ L ( 1 ) a a a

where d is the distance between P1 and P2, so that

= V(x2 - ^)2 + (y2 - y i f + (z2 - zxf (2)

For any set of direction cosines, cos2 oc + cos2 /? + cos2 y — 1.


Va* + b* + c* r Va* + b2 + c2 ^ ( 3 )

cos y = . Va* + b* + c*

The angle 6 between two lines with direction angles ocv /3ly y1 and a 2 , j82> 72

may be found from

cos 0 — cos cos a 2 + cos /?x cos j82 + cos y x cos y2 (4)

The equation of a plane is ^ + By + = Z). A, B, C- are direction ratios for any normal line perpendicular to the plane. The distance from the plane to Px = (xvy^z^ is '

Ax1 + Byl + Cz1-D ( 5 )

VA* + B* + C 2

The equations of a straight line with direction ratios A, B, C through the point P1 = x^y^z^ are

x — x x y —y1 z—z1. ( f . A B C K

The angle 9 between two planes whose normals have direction ratios] Alf

Bl9 C± and A2y B2, C 2 , or between two lines with these direction ratios may be found from

a 4*A* + B1BA±C1C1

cos u — —. - -— . = (7)

VA^ + B* + Q 2 V ^ 22 + B* + C 2

2 1 ;

2.7. Plane right triangle. C = 90°, opposite c.

c= Va2 + b\ A = tan- 1 — a > (1)

b = c cos .4, a — c sin 4 ;

2.8. Amplitude and phase

a cos to* + ^ sin a)t = c sin (otf + A) (1)

where the amplitude c and />A*we 4 are related to the constants a and b by the equations of §2.7.

2.9. Plane oblique triangle. Sides a, by c opposite A, By C.

A + B + C = 180 degrees or 7r radians

If a, b, c are direction ratios for a line, or numbers proportional to the direction cosines, then

a Q b COS p :

BASIC MATHEMATICAL FORMULAS § 2.10

Law of sines : ——v = ——— = — — ~ (1) J sin A sin B sin C w

Law of cosines : c1 = a2 + b2 — 2a£ cos C (2)

Area i£ = \ab sin C = —d)s — b) (s — c) (3)

where s — \a + b + c).

2.10. Spherical right triangle. C = 90°, opposite c.

sin a = sin A sin c, sin b = sin 5 sin c (1)

sin a = tan b cot sin b = tan a cot A (2)

cos 4 = cos a sin 5 , cos B = cos 6 sin A (3)

cos 4 = tan b cot c, cos B — tan a cot c (4)

cos £ = cot A cot JB, cos c = cos a cos 6 (5)

2.11. Spherical oblique triangle. Sides a, b, c opposites A, B, C.

0° <a + b + c < 360°, 180° < A + B + C < 540°

Law o/ : sin a sin 6 sin A sin JB sin C

Law of cosines:

(i)

cos c = cos a cos b 4- sin a sin 6 cos C (2)

cos C = -cos A cos JB + sin A sin JB cos c (3)

Spherical excess: E = A + B + C-\m°

If JR is the radius of the sphere upon which the triangle lies, its area ^ = 7 7 ^ / 1 8 0 . If

s = i(a + b + c),

tan — = Vtan %s tan ^(s — a) tan ^(s —b) tan \s — c) (4)

2.12. Hyperbolic functions. For a single number x, the equations g—X gX _j_ £— X

sinh a: = , cosh x = - (1)

§ 2 . 1 3 BASIC MATHEMATICAL FORMULAS 9

define the hyperbolic sine and hyperbolic cosine in terms of the exponential function. And the equations

. sinh x , cosh x \ tanh x = : — , coth x = -—, j cosh x sinh x f . 1 1 < ^ ' sech x = — , csch x •

cosh a:' sinh x

define the hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant, respectively, in terms of the hyperbolic sine and hyperbolic cosine. The four functions sinh x, tanh x, coth xy and csch x are odd functions, so that sinh (—x) = —sinh x and tanh (—x) = —tanh x. The functions cosh x and sech x are even, so that cosh (—x) = cosh x. Between the six hyperbolic functions we have the relations

cosh2 x — sinh2 x = 1, coth x — ^ tanh x > (3)

sech2 x + tanh2 x = 1, coth2 x — csch2 x = 1 )

2.13. Functions of sums and differences

sinh (x + y) = sinh x cosh y -f- cosh x sinh y sinh (x —y) = sinh x cosh — cosh x sinh y cosh (x -\- y) = cosh a? cosh j ; + sinh x sinh _y cosh (x — y) = cosh x cosh j — sinh x sinh j

(i)

2.14. Multiple arguments

sinh 2x = 2 sinh a? cosh cosh 2# = cosh2 x -f- sinh2 a? (1)

sinh2 x = \ cosh 2x — -J, cosh2 x = ^ cosh 2# + (2)

2.15. Sine, cosine, and complex exponential function, i2 = — 1.

£ia: __ c o s ^ _|_ j s in g - fo _ C os a? — z sin X (1) gicc _|_ ix gix g~ix

cos x = _ — , sin x = (2) 2 2z

2.16. Trigonometric and hyperbolic functions

cos ix = cosh cosh ix = cos a? sin ix = i sinh a?, sinh ix = i sin x J> (1) tan za? = z tanh a?, tanh ix = i tan a?.


The wth derivative is

d

dx''

3.3. Partial derivatives. Let # = f(x,y) be a function of two variables. Then the partial derivative dz/dx is defined by the equation

dz f(x + Ax,y) —f(oc,y) dz (1)

2* 17. Sine and cosine of complex arguments

sin (x + iy) = sin x coshjy + i cos x sinh y (1)

cos (x - j - iy) = cos x cosh y —i sin # sinh y (2)

2.18. Inverse functions and logarithms

c o s - 1 x = —z ln(# + z'Vl —#2)> c o s h - 1 # = ln(# + V # 2 — 1) (1)

s i n - 1 # = —i ln(ix + Vl — # 2 ) , s inh - 1 a? = ln(# + ^/x2 + 1) (2)

x t 1 + «c , , 1 , 1 + # / o x t a n - 1 x = In — , tanh - 1 * = — In , (3) 2 1 — ix 2 I —x v /

3. Differential Calculus

3.1. The derivative. Let y =f(x) be a function of x. Then the derivative dyjdx is defined by the equation

< t = H m Ay = i i m = / ( * + * * ) - / ( « ) Ax-Ô t\X &x-^0 I\X

Alternative notations for the derivative are /'(#)> Dxy, y\ and j ; is used for dyjdt, when y = F(t).

3.2. Higher derivatives. The second derivative is

Similarly the derivative dz/dy is formed with y varying and x fixed.


The second partial derivatives are defined by

d2z _ _d_(d*\ dx2~dx\dx)i

a 2 £___a_/a£\ d2z _ d /dz\ " ~ 1 /' dy2~ dy\dy)'

(2)

~VJ dy\dx/ dx dy dx\dy) dy\dx/ dy dx

The same process defines partial derivatives of any order, and when the highest derivatives involved are continuous, the result is independent of the order in which the differentiations are performed.

For a function of more than two variables, we may form partial derivatives by varying the independent quantities one at a time. Thus for u = f(x,y,z) the first derivatives are du/dx, du/dy, dujdz.

3.4. Derivatives of functions

Inverse functions: dy 1 d2y — d2x\dy2

y=y(x), x = x(y), - = _ = (1)

™ . T r/ \ / x dy dy du Cham rule: y = / ( « ) , « = * ( * ) , - = ^ (2)

1 dv Dfldx

Implicit function: f(x9y)=Q9 ~£ = ~dffd^ ^

Parameter: y = y(t)9 x — x(t) dy . dx ...

L e t y = *> x=Tt ^

Then dy y d2y _ xy —yx dx x' dx2 xz

r • d . , 7 . du , , dv ' ,e\ Ltneanty: - an + bv) = a - + b ^ (5)

3.5. Products

(uv)' — uv' + (uv)" = uv" + 2wV + vu" (1)

Leibniz rule :

(uv)W = uvnr+ » C 1 « ' © ( n - 1 ) + ... + n C> ( r ) z>< w - r ) + ... + uMv (2) The n C r are defined in §1.3.

12 BASIC MATHEMATICAL FORMULAS §3.6

3.6. Powers and quotients

, w ', . / 1 Y , 1 W —v' (u \' vu' —uv' ( * ) ' = » * - V , ( - j = ( 0 ' = - 5 - , ( - ) = — ^ —

3.7. Logarithmic differentiation <*(lnjQ = /

(i)

v = — , W ^ - ^ - ' ( T + T - J ) <•> wa?;& , J(ln v) i( u' _ a>'\

(2)

y = uv

3.8. Polynomials

= In « ) ' = + *>' In u j (3)

—(anxn + ... + a r * r + ... + a±x + a0) = nanxn~x + ... + rarxr~x + ... + a± (2)

3.9. Exponentials and logarithms. We write In for loge, where ^ = 2.71828, as in §1.2.

nx J„x (1)

(2)

dex dax

= (In a)ax

dx dx = (In a)ax

d\nx 1 d logr t x dx x9 dx X

3.10. Trigonometric functions

. d 0 d / i X — sin # = cos — tan x = see* — sec # = tan x sec # (1) dx dx dx

d . d 0 d — cos x = —sin a?, — cot x = — esc2, x, — esc # = —cot x esc x (2) dx dx dx

3.11. Inverse trigonometric functions d . , 1


3.12. Hyperbolic functions

^ sinh x = cosh x, ^ tanh x = sech2 xf

sech x = —tanh x sech x ax

cosh x = sinh a:, coth a? = —csch2

dx dx

~ csch # = —coth x csch x dx

(i)

(2)

3.13. Inverse hyperbolic functions

^sinh_1* = vF+T s^_ 1* = r=* ( 1 )

^ c o s h "* = ^tr ' ^ c o t h " l - 1 J = ^ L T ^

3.14. Differential. Let y = / ( # ) , and A# be the increment in x. Then

= Ax and dy =f'(x)dx = ^¥^dx (1) Parameter:

y = * = *(0> dx = x dt, dy =y dt (2) First differentials are independent of the choice of independent variable.

3.15. Total differential. For two independent variables z=f(x,y\

7 I^jst _ , _ r dz dz dx , dz dy / 1 X

^ = ^ ^ + ^ ^ a n d di = 8xdi + dydi &

For three variables, u =f(x,y,z),

d u = ^ d x + 8^dy+^dz ( 2 )

du du dx du dy du dz du du dx du dy du dz . ^ = l^dt+dydi + dzdi' dv^ dxdv +dydi + -dv ^ >

and similarly for functions of more variables.

3.16. Exact differential. The condition that as in §3.15 for some f(x>y)>

A(x,y)dx + B(x,y)dy = dz, with z = f(x.y) is

dA^dB^ dy ~ dx ^ '


for some f(xyyyz) to make

A(xyyyz)dx + B(xyyyz)dy + Cxyyyz)dz = du,

with u = f(xyyyz)y the condition is

3C___dB_ 3A__dC_ 8B _ 8 A dy dz' dz dx' dx dy (2)

3 .17. Maximum and minimum values. Lety = f(x) be regular in the interval ay b. Then at a relative maximum, / ( % ) with a < xx < 6, / ' ( # ) decreases from plus to minus as # increases through x±. At a relative minimum, f(x2) with a < x2 < byf\x) increases from minus to plus as x increases through x2. Thus the largest and smallest values of f(x) will be included in the set / («) , / (£) , andf(xk), wheref'(xk) = 0.

3 .18. Points of inflection. The graph of y = f(x) is concave downward in any interval throughout which f"(x) is negative. The graph is concave upward in any interval throughout which f"(x) is positive. At any point y3 =f(x3) such thatf"(x) changes sign as x increases through x.s, the graph of y =f(x) is said to have a point of inflection.

3 .19. Increasing absolute value. Let OP = s(t)y the distance along OX. Then the velocity v = ds/dt = sy and the acceleration a = dvjdt = v = d2sjdt2 = s. Then 5 increases when v is positive, and v. increases when a is positive. The distance from O, or \s\ increases when |s|2 = s2 increases, so that \s\ increases when ss is positive, or when s and v have the same sign. The speed or \v\ increases when \v\2 = v2 increases, so that \v\ increases when vv is positive, or when v and a have the same sign.

3 .20. Arc length. In the triangle formed by dx, dyy and ds, the differential of arc length, the angle opposite dy is the slope angle, and the angle opposite ds is 90°. Thus

dy dx == ds cos t , dy = ds sin ry -j- = tan r, and ds2 = dx2 + dy2, (1)

ds = V I + y'2, dx = \/x2 + y2 dt (2)

In polar coordinates, r = r(6)y

ds2 = t/r2 + r2dd2

and ^ = V r ' 2 + r*d6= V > 2 + r 2 ^ ^ (3)


a = V * 2 + j ' 2 = aJ + ^ T (1)

3.23. Theorem of the mean. Let f(x) and be regular in the interval a> b. Then for at least one value xk with a < xk < 6,

Rollers theorem:

If / ( a ) = 0 , / (6) = 0, /'(*0 = Q (1)

Law 0/ mean :

f(b)=f(a) + (b-a)f'(x2) (2)

Cauchy's mean value theorem: If F'(x) ^ 0 inside a, b, (3)

- / ( a ) = / ' ( * , )

3.24. Indeterminate forms. Let f(x) and F(x) each approach zero as x approaches a. We briefly describe the evaluation of lim f(x)/F(x) as the indeterminate form 0/0. When f(x) and F(x) are each analytic at a, and series in terms of powers of (x — a) can be easily found, the limit may be found by using these series.

VHospital's rule: If the limit on the right exists, then when /(#)—>-0, — 0 ,

lim = hm - j ^ r r (1) x-â F(x) x-+a F (X) V

3.21. Curvature. R = dsjdr is the radius of curvature, and the curvature K = \/R is given by

K =l = y" = *y —y* m

R \+y'*fl* (*2 + v 2 ) 3 / 2 l )

In polar coordinates r = r(9), r2 + 2r'2 -rr"

3.22. Acceleration in plane motion. The velocity vector has x and y components x = dxjdt and y = dy/dt, and magnitude v = Vx2 + y2. The acceleration vector has x and y components x = d2xjdt2 and y = d2y/dt2. It has a tangential component v = dvjdt and a normal component v2jR. As in §3.21, IjR = The magnitude of the acceleration is

16 BASIC MATHEMATICAL FORMULAS § 3 . 2 5

In this rule x may approach a from one side, x —a-\- or x —> a—, and it applies when in place of x—> a, x—> + <», ox x — > — <x>. The rule in any form also applies when as x -> a, f(x) and F(x) each tend to infinity. This is the indeterminate from 00/ 00.

By writing / = 1/(1//) for one of the factors, we may reduce the form 0-°o

to 0/0 or 00/ 00 and then use I'Hospital's rule. A similar procedure with each term sometimes reduces the form <x> — 00 to 0/0.

If the evaluation of l i m / F leads to an indeterminate form 0°, 10 0, c o ° , the evaluation of L = l imJ^ln/ leads to a form 0 • » or 00 • 0. This may be found as indicated above, and then lim fF = eL.

3*25. Taylor's theorem. Let f(x) be analytic at a. Then

fix) = / ( a ) +f(a)^Jf^ + / > ) ( ^ _ f ) ? + / ' > 0 i _ f ) !

+ . . . + / < » - i > ( 8 ) ( * - a ) ^ V

(» - 1)! For real values, the remainder after the term with /^ n _ 1 ) ( t f ) is

(1)

( 2 )

where xx is a suitably chosen value in the interval a, x. An alternative form is

f(a + h) =fa) + / » ^ - +/'"(<>%• + - + / ( n , ( « ) J f + - ( 3 )

The special case when a = 0 is called Maclaurin's series.

f(x) = / ( 0 ) + / ' ( 0 > £ + / " ( 0 ) J + / " ' ( 0 ) J + . . . + / " " ( o g (4)

For computation these series are usually used with (x — a) or h small, and the remainder has the order of magnitude of the first term neglected.

Let f(x,y) be analytic at (a,b). Then in the notation of §3.3,

A*>y) =M*>) + (x- a)fx(a,b) + (y- b)fy(a,b)

+ ^[(* - *)*fM) + 2(x -a)y- b)fxyayb) + (y - & ) 2 A „ ( a , * ) ]

(x — a) dX

Y=b

(5)


An alternative form is

f(a + h,y + k)= f(a,b) + hfx(a,b) + kfyayb)

+ ... 1 / d 3 \n

*.h^+kWf(x'y) y— b

The special case when a = 0, b = 0, Maclaurin's series is

f(x,y) = / ( 0 , 0 ) + */B(0,0) + yM0,0) + ...

1/8 8\n

\x=0 \y=0

+ -And similar expansions hold for any number of variables.

3*26. Differentiation of integrals

d rx 7 . „ • d rb d_ dx

^fx,t)dt=fx,x)+l^xfx>t)dt

If f(x,x) is infinite or otherwise singular one may use

dt

d fv(x)

dx jVu(x/^dt = v'(x)f[x>v(x)] - w ' W / ^ ^ W l + J " * * j ^ f ( x ^ d t

4. Integral Calculus

( 6 )

(7)

(1)

(2)

(3)

(4)

(5)

4.1. Indefinite integral. With respect to xy the indefinite integral of f(x) is F(x) provided that dFjdx = F'(x) —f(x). The indefinite integral of the differential f(x)dx is F(x) if dF(x) = f(x)dx.

Jf(x)dx = F(x) if F\x)=f(x) or dF(x)=f(x)dx (1)

For any constant C, .F(#) + C is also a possible indefinite integral. By using all values of C, and any one F(x)y we obtain all possible indefinite integrals :

$f(x)dx = F(x) + C.


J +•. . .+ # r # r + ... + # + a0)dx

1 - + . . . + —|~r flr*r+i + . . . JL ^ 2 + ^ + C

(2)

~ n + 1 n 1 1 r + 1 r ~ 1 2

4.4. Simple rational fractions

\1ArrdX = Aln\X-r\ + C, j ^ A ^ ^ A ^ ^ + C ( 1 )

f 2 ^ ( « - a ) - 2 g f t < f a = 2 R e f ^ 4 + B t _ < f a ) J (x — a)2 + b2 J x — a — bi ^ ^

- ^ l n [ ( ^ - « ) 2 + 6 2 ] - 2 J B t a n - 1 ^ ^ + C J

— 2 = — tan 1 f -C, — 5- = — In —,— + C (3) J a?2 + a a a J x* — a* 2a \ x + a I

J^ = l n | * | + C (4)

4.5. Rational functions. T o integrate R(x)dx where R(x) is a rational function of x, first write i?(#) as the quotient of two polynomials P1(x)jD(x).

4.2. Indefinite integrals of functions

jdF(x) = F(x) + C, djf(x)dx = f(x)dx (1)

Linearity:

j(au + hv)dx = a$u dx^ b$v dx (a and b not both zero) (2)

Integration by parts:

J u dv = uv — J* v du (3)

J u^dx — uv — J v^dx (4)

substitution i \^y)dx = \^y)%dy = \^dxdy ( 5 )

4.3. Polynomials

Jo<fo = C, j a dx = ax + C, j £ww ^ = —J - j (1)


If the degree of P^x) is the same or greater than the degree of D(x), by division find polynomials Q(x) and Px) such that

lRWx = \ - 0 ) d x = i Q x ) d x + i w ) d x ( 1 )

where Px)jD(x) is a proper fraction, that is, has the degree of P(x) less than the degree of D(x). Integrate Q(x)dx as in § 4.3.

T o integrate the proper fraction P(x)jD(x), decompose it into partial fractions as follows. Suppose first that the roots of D(x) = 0 are all distinct. Let D(x) be of the nth degree and call the roots r v r2, rn. Then

D(x) - x - r± + Y~r~2

+ " ' + x - rn W l t h A k " D\rk) 2 )

Assume that P(x) and D(x) have real coefficients. Then either r and A are real and we use the first equation of § 4.4, or complex roots occur in conjugate pairs, a + bi, a — biy with conjugate numerators A + Bi, A — Biy and we use the third equation (2) of § 4.4.

If the equation D(x) = 0 has multiple roots, each factor (x — r)m in D(x) will lead to a series of fractions

P(x) P(x) A' A" | A™ D(x) (x - r^D^x) (x-ry (x - rf ^ ''' ^ (x - r)m ^ '" v ;

T o determine the m constants, solve the system of equations

P(r) = A^D^r), P\r) = A^D^r) +'^<™-i> (4)

P"(r) = A™D^\r) + lA^-UD^r) + 2 - M ( w - 2 ) 2 ) 1 ( r ) , ... (5)

p < m - D ( r ) = A^D^-^r) + (m — \)A^~^D^r) \

+ m-\)m~ 2)Aî)D1,,(r) (6)

+ ... + (m-1)1 A'D^^r) )

To integrate the simple fractions, use the first and second equations (1) of § 4.4. Conjugate complex roots will lead to conjugate fractions which may be combined in pairs to give a real result after integration.

4.6. Trigonometric functions

j sin ax dx = ^- cos ax + C, j sin2 ax dx = — s*n^**x _|_ q (1)

j cos ax dx — ^ sin ax + C, j cos 2 ax dx = + s*n^*lx _j_ q (2)

20 BASIC MATHEMATICAL FORMULAS

4.7. Exponential and hyperbolic functions

f eax dx = —eax + C, [ bax dx = — ^ — r Z>a* + C J a J a In 0

sinh ax ax = — cosh ax + C, f cosh ax ax = — sinh ax + J a J a

4.8. Radicals

J v A " ^ ( t ) + c = - -~ - 1 ( t ) + c^

f Va^^c^dx = 4 - V^Zrx* + °~ s in- 1 — + C J 2 2 a

J -\A2 + 4 ax = y V*2 + -4 + y In I x + V x 2 + 4 ! * + C

* For A positive, A = a2, In (x + -y/V2 + a2) = sinh - 1 #/a + In a

For 4 negative, A = —a 2 , In -f- -yA2 — a2) = cosh - 1 xja + In a

In each case In a or (-4/2) In « may be combined with the constant C.

f dx r _ 1 , 1 ax\ ' -— = csc ax dx = — In tan — I + C J sin ax J a \ 2 I

= — In I csc ax — cot ax I + C a 1 1

f dx r , 1 , ! lax , 77 \ 1 , „ = sec ax dx = — In I tan I — + — I \ + C J cos ax J a \ \ 2 4 / 1

= — In \ sec ax + tan ax I + C a 1 1

f . ^ — = [ csc 2 ax dx = — cot ax + C J smzax J a

J Ltan ax dx = In | cos ax | + C

I —-7T— = f sec2 ax dx — — tan ax + C J cos"5 ax J a

f cot ax a*x = — In I sin ax I + C J a 1


4.9. Products

j eax sin bx dx = ^ 2 ^ ^2 £a a ;(a sin bx — b cos 6x) + C (1)

r l J eax cos 6x dx = ^ 2 ^2 ea a ; (a cos bx -\- b sin 6x) + C (2)

J xea* dx = ^ eax(ax — 1) + C, J In ax ax = x In ax — x + C (3)

We may use § 2.15 to express factors like sin ax, cos bx in terms of complex exponentials. And we may use § 2.12 to express factors like sinh ax, cosh bx in terms of exponentials. Thus any product of such factors, or product times ekx, may be reduced to a sum of exponential terms like that in the first equation of § 4.7. If powers of x are also present we are led to

J xneax dx = — xneax — ^ J xn~xeax dx (4)

This reduction formula may be used repeatedly until the power of x disappears when n is a positive integer.

Let / (x ) be the natural logarithm, an inverse trigonometric function, or an inverse hyperbolic function of some simple function of x, for example, In (ax + b), s i n - 1 x, tanh - 1 x2. Then for n = 0, 1,2, ... the equation

f xnf(x)dx = —\-j x ^ x ) -J-j- \ xn^f(x)dx (5) J 71 ~j~ 1 71 - j - 1 J

often leads to a simpler integral.

4.10. Trigonometric or exponential integrands. Any rational function of trigonometric functions of x may be written in the form R(sin x, cos x), and the substitution t = tan x/2 makes

£(sin x, cos x)dx = RI t + f 2 , yj^rJ Y + l 2 ^

The new integrand is a rational function of t, discussed in §4.5. If the integrand is a rational function of eax, let t = eax and

R(eax)dx = R(t) ^ A (2)

This may be applied to i?(sin x, cos x) with a = i, if § 2.15 is used.


4 4 1 . Algebraic integrands. L e t p and q be integers and a ^0; then the substitution t = (ax* + b)l/q makes

R[x, (ax + b)*l*\dx = R(^^~> tp)J X t*-1 dt (1)

The new integrand is a rational function of t, discussed in § 4.5. For certain integrands of the form R(x, /Y/px2 + qx + r) the following

transformations am useful.

1 makes dx dy

x \/px2 + qx + r \/ry2 -\- qy -\- p

—— makes

dx dy

(x — 5) s/px2 + + r V(ps2 + <?s + y*) 2 + (2ps + <?);y + p

y = x + X makes 2£

(2)

(3)

!?(*, V^ 2 + + r)dx = R ^ y - ± , V ^ 2 + ^4p^)dy W

Simple integrals of this type are given in § 4.8. Also note that

R(x, V # 2 — # 2 ) ^ = sin <z cos t)a cos £ aft if a? = a sin £ (5)

R(xy V # 2 — ^ = sec a tan t)a tan £ sec t dt if # = a sec £ (6)

\/x2 + A 2 ) ^ = tan # sec t)a sec2 £ ^ if x = a tan £ (7)

The new integrands in t are essentially those treated in § 4.10.

4.12. Definite integral. Let a , x2i ... xn_x be points of subdivision of the interval ayb such that xk < xk+1, with a = x 0 , b = xn. Let fc = xk ~ xk-i> a n d dM be the maximum value of 8k. Then

- fimJM1)S1+M2)S2+ ...+ADK] (i)

For each choice of and the the £ k are any values such that Xjc-i £ki^ xk. The limit exists for f(x) regular on ay b :

j f(x)dx = j f(x)dx b = F(x)\b = F(b)-F(a) (2)

where F(x) is any function whose derivative with respect to x is f(x), as in §4.1.


4.13. Approximation rules. Let the interval ayb be divided into n equal parts each of length h so that b — a = nh. And let f(a + kh) = yk ; £ = 0, 1,2, Then the trapezoidal rule is

\bJx)dx = •^y1 + 2y2 + 2y3+ ... + 2yn-1 + yn) — Rn (1)

where Rn = ^(b — a)h2f"(x1) (2)

for some suitable x± in a,b.

The more accurate Simpson's rule requires that n be even, and is

jbJ(x)dx = ^(yi + 4y 2 + 2jy3 + 4y 4 *

+ ... + 2 j n _ 2 + 4y n _ ! + ;y„) - Rn

(3)

where i?„ (b - «) /*y I V (* 2 ) (4)

for some suitable x2 in ayb.

In Gauss' Method we let x = \ (a + b) ^(b — a)w. Then

Fbf(x)dx = \1 Ô)^ = Y Rmg(™m) (5)

The wm are the w roots of

Pn(x) = 0 (6)

(see § 8.2), and for r = 0, 1, 2, ... w — 1.

In particular,

n = 2: w1 = -w2 = 0.57735, R± = R2= |, £ (x 4 ) =

= 3 : w1 = —w3 = 0.77460, ^ 2 = 0> R± = Rz = R2 = ^, E(x6) =

n = A: w1 = -w± = 0.86114, w2 = ~ws = 0.33998, ^ = #4 = 0.17393, ^ = ^3 = 0.32607, ^ 8 ) = T ^ f 2 - 5

The formula is exact if f(x) is a polynomial of degree not exceeding 2n — 1. The error for a higher degree polynomial may be estimated from the given value of

1 n A 1 J 1 m=l


4.14. Linearity properties. Linearity in the integrand:

[b [Af(x) + Bg(x)]dx = A [bfx)dx + B fb g(x)dx J a J a J a

Linearity with respect to the interval:

jb f(x)dx = — j^f(x)dx

I bf(x)dx = [C f(x)dx + f bf(x)dx J a J a J c

4.15. Mean values. The mean value of f(x) on the interval afi is

^ = b ^ \ b J ^ d x

If f(x) is continuous on afi then for some xx on the mean value

/ = f(xi) o r jb f(x)dx = (b — a)f(x^) mean value theorem

The root-mean-square value off(x) on the interval afi is

(i)

4.16. Inequalities. Assume that a <b. I f / (# ) < in then

f 6 / (#)^# < \b g(x)dx J a J a

If m <f(x) < M in ayb then

m(b — A) < | bf(x)dx < Af(fc — a) J a

(2)

(l)

(2)

(3)

(1)

(2)

If | f(x) | M in # ( o r on the complex path of integration C of length L) , then

| | < M(b - a) or | \cfz)dz | ^ M L

Schwarz's inequality:

With the notation of §4.15, and


4.17. Improper integrals

j"f(x)dX = lim f'jWdx \

> (1) jb_J(x)dx = Umjb_tf(x)dx )

j2nf(x)dx = jc_xf(x)dx + j™f(x)dx (2)

If / ( * ) becomes infinite or has a singularity at x = b with b â,

\[fx)dx = \\m^hfx)dx (3)

4.18. Definite integrals of functions. Let u(x) and z;(x) be continuous in a,b. Then

j b du(x) = u(b) — u(d) (1)

f& u(x)v'(x)dx = w(#W#) 6 — f6 v(x)u'(x)dx (2)

Let £ = a=.g(c), b = g(d)y and ^'(w) maintain its sign in c,d. Then

J'/(0^= \dJ[gu)]g»du (3)

4.19. Plane area

A = j / r dr d8 = d6 J-;:; , * = i- ( V - r l V » (2)

4.20. Length of arc. See § 3.20. For a plane curve,

s = f*2 Vl + / 2 = fh V ^ + J 2 A ( 1 )

5 = I'"2 V ' 2 + r 2 dd = f ' 2 V>2 + r2Q2 dt (2)

4.21. Volumes. With suitable limits on the iterated and triple integrals,

V = r 2 A(x)dx = $dx$[z2(x9y) — z^xy^dy = $dd$(z2 — zj r dr (1)

V = J J J <fy dz = JJJ r <fr <W <fe = JJJ r 2 sin <f, dd dcj> dr (2)


4.22. Curves and surfaces in space

s = P2 v* 2 +y 2 + z2 dt (i)

where x = x(t)y y = y(t)y z = z(t).

A = jj VfJ+fS+1 dxdy (2)

where z = f(xyy).

4.23. Change of variables in multiple integrals. If

F[x(uyv)y y(u,v)] = G(u,v) (1)

and the limits are suitably related, then

J j F(x,y)dx dy = J J* G(uyv) du dv (2)

Similarly if F[x(uyvyw)y y(uyvyw)y z(uyvyw)] = G(uyvyw) (3)

then

HI F(xyyyz)dx dydz= j J J G(uyvyw) du dv dw (4)

The Jacobians inserted here are I xu Ju Zu

8(x>y) _ I *« ^ « I - - « _ - v 8 ( ^ ^ ' g )

S ( H , » ) _ U „ j J u y v v y u ' d(u,v,w) X V y V % V (5)

If the Jacobian determinant is not zero, the transformation x = x(uyv)y

y = y(uyv) has an inverse transformation. When there is a functional relation

F[x(uyv)yy(uyv)]=0 (6)

the Jacobian d(xyy)jd(uyv) is identically zero. Also

d(xyy) 8(XyY) = d(xyy) 3(xyy) 1 3(XyY) 8(uyv) d(u,v)' d(uyv) d(u9v)/d(x,y)

Similar results hold for any number of variables.

4.24. Mass and density. Let p be the variable (or constant) density of a curve, area, or volume. Then the total mass is given by

M = I dm


where dm = p ds, dm = p dA, or dm = p dV. For the differentials ds, dA, dV, use the integrands from §§4.19-4.22. Mean density is

- fpds _ JpdA . jpdV 9 \ds' 9 jdA' 9 )dV

4.25. Moment and center of gravity. Let x, y, z be the coordinates of the center of gravity of one of the elements of mass dm of § 4.24. Then the moments about the coordinate planes are

Myz = Jx dm, Mzx = jy dm, Mxy = jz dm (1)

And the center of gravity of M is x, y, z where \x dm (y dm fz dm .

* = - t t - > y = Jn-> 2 = (T~ 2 )

J dm t J dm jdm For a mass M composed of several, e.g., three, parts Af1 ? M2i M a , the masses

and moments are additive so that

X M M± + M2 + M3 [ )

and similarly for y and z.

4.26. Moment of inertia and radius of gyration. Let x> y, z represent root-mean-square values of coordinates for any one of the elements of mass dm of § 4.24. Then the second moments about the coordinate planes are

Iyz = JV2 dm, Izx = J>2 dm, Ixy = JV2 dm (1) The moments of inertia about the coordinate axes are

/, = /„ + !„,= S(y* + z*)dm (2)

and similarly for Iy and Iz. For a figure in the xy plane, z = 0, Ixy = 0, and if Iz = I0,

I0 = IX + IV = f(x*+y*)dm= fr*dm (3)

For any moment of inertia, as Ix, the corresponding radius of gyration k is defined by

Af*« = / „ or k=VlJM (4) For a mass M composed of several, e.g., three, parts Ml9 M2, M.6, the

masses and moments of inertia are additive, so that

28 BASIC MATHEMATICAL FORMULAS §5-1

If I is the moment of inertia of M about any axis, and Ig is the moment of inertia of M about a parallel axis through the center of gravity, and the distance between the parallel axes is dy then

I = Ig + Md* (6)

5. Differential Equations

5.1. Classification. Any equation that involves differentials or derivatives is a differential equation. If the equation contains any partial derivatives, it is a partial differential equation. If it does not, it is an ordinary differential equation.

The order of the differential equation is the same as the order of the derivative of highest order in the equation.

Suppose that a differential equation is reducible to a form in which each member is a polynomial in all the derivatives that occur. Then the degree of the equation is the largest exponent of the highest derivative in the reduced form.

A differential equation is linear if it is a first-degree algebraic equation in the set of variables made up of the dependent variables together with all of their derivatives.

5.2. Solutions. Consider a single ordinary differential equation with dependent variable y and independent variable x. Then y = f(x) is an explicit solution of the differential equation if this equation is identically satisfied in x when we substitute / (# ) , f'(x), e tc for yy dyjdxy etc therein. Any implicit relation F(x,y) = 0 is a solution if when solved for y it leads to explicit solutions. Every ordinary differential equation of the nth order admits of a general solution containing n independent constants.

5.3. First-order and first degree. Consider an ordinary differential equation of the first order and first degree. Any such equation may be written in one of the forms

M<bc + Ndy = 0, M+N% = 0, or | = - f (1)

where M(x,y) and N(x,y) are functions of a?'and y. We indicate how to recognize and solve certain special types in §§ 5.4-5.8.

5.4. Variables separable. Here the M and N in § 5.3 are products of factors, where each factor is either a function of x alone, or a function of y


alone. Divide the equation by the factor of M containing y and by the factor of N containing x. Then the differential equation is

f(x)dx = gy)dy (1)

in which the variables are separated. The solution is

jf(x)dx = J gy)dy + c or j * f(x)dx = j ^ g(y)dy (2)

if -y = yQ when a? = x0.

5.5. Linear in J>. Here AT" in § 5.3 is a function of x alone, and M is a first-degree polynomial in y. Thus the equation is

A(x)^ + B(x)r=C(x) or g + P ( % = 0(«) (1)

after division by A(x) and with a new notation. Calculate

[P(x)dx and / (* ) = e J P d a j (2)

with any constant in the integral. The solution of the equation is

y^^pi^QWdx + c] (3)

If y = yQ when # = x0, the solution is found from

I(x)y-I(x0)y0= I(x)Q(x)dx (4)

5.6. Reducible to linear. T o solve the Bernoulli equation

A(x) ^ + B(x)y = C(x)yn, n^l or 0 (1)

multiply by (1 — n)y~njA(x) and make the substitution

The resulting equation is like that of § 5.5, with u in place of v. If M in § 5.3 is a function of y alone, and M is a first-degree polynomial

in Xj the differential equation is

A(y)fy + By)x=C(y) (3)

and may be solved as in § 5.5 with the roles of x andjy interchanged.


Whenever we observe new variables u(x,y) and t(x,y) which reduce a given differential equation to the form

Ai)d^t+Bt)u = Ci) (4)

we may solve as in § 5.5 with t and u in the roles of x and y.

5.7. Homogeneous. Here the M and N in § 5.3 are each homogeneous of the same degree, that is of the same dimension when x and y are each assigned the dimension one. In this case use

y = vx, dy = v dx + x dv (1)

to eliminate y and dy. The new equation in x and v will be separable as in § 5.4. If more convenient, let x = uy.

5.S. Exact equations. The equation of § 5.3 will be exact if

dMjdy = dNpx

Integrate M dx, regarding y as constant and adding an unknown function of y, say/(j>). Differentiate the result with respect to y and equate the new result to N. From the resulting equation determine the unknown function of y. The solution is then

jMdx+f(y) + c = 0 (1)

If more convenient, interchange M and N and also x and y in the above process.

5.9. First order and higher degree. The general differential equation of the first order is

if we write a single letter p in place of dyjdx. For special solvable types see §§5.10-5.11.

5.10. Equations solvable for p. Some equations §5.9 may be easily solved for p. The resulting first-degree equations may often be integrated by the methods of §§ 5.3-5.8.

5.11. Clairaut's form. Clairaut's equation is

§5.12 BASIC MATHEMATICAL FORMULAS . 31

and dx=P' lx*=dx ( )

= o b e c o m e s F*>p£)=° ( 6 )

The solution of this first-order equation in the variables x and p may be found as in § 5.3 or § 5.9. Suppose it is written in the form/) = G(x,c1); then

y = jG(xic1)dx + c2 (7)

is the general solution of the second-order equation with y missing.

If the letter x is missing, we put

d y * d*y * d p m

and

411)=° bec°mes 4 # l ) = 0 (9)

The solution of this first-order equation in the variables y and p may be found as in § 5.3 and § 5.9. Suppose it is written in the form/) = G(y,c1); then

* = J ^ 3 + t f - ( 1 0 )

is the general solution of the second-order equation with x missing.

where / is any function of one variable. The general solution is

y = cx+f(c) (2)

This represents a family of straight lines. They usually have an envelope given in terms of the parameter c by

* = -/'(<). y=fc)-cf'c) (3)

These are the parametric equations of the singular solution.

5.12. Second order. The general differential equation of the second order is

If either of the letters x or y is absent from the function F, the substitution dyjdx = p reduces the equation to one of the first order. If the letter y is missing, we put

dy d2y dp


5.13. Linear equations. The general linear differential equation of the second order is

Ax) % + Bx) %+Cx)y = Ex)' w i t h A~x) *0 ( 1 )

We get the corresponding homogeneous equation by replacing E(x) by zero. Two solutions uv u2 of the homogeneous equation are linearly independent and constitute a fundamental system if their Wronskian determinant W(x) is not zero, where

% u2

uJ Uc, W(x) = uxu2 — u2Ui = (2)

When W(x) ^ 0, u± and u2 constitute a fundamental system of solutions of the homogeneous equation. And if u is any particular solution of the non-homogeneous equation with E(x) present, the general solution of this equation is given by

y = u + cxux + c2u2 ( 3 )

In this form c± and c2 are arbitrary constants, u is called the particular integral, and cxux + c2u2 is the complementary function.

Similarly for the linear equation of the nth order

Alx)Âlx)y = Ex) (4)

the general solution is

y = u + cxuY + c2u2 + + cnun (5)

Here w, the particular integral, is any solution of the given equation, and uv w2, un form a fundamental system of solutions of the homogeneous equation obtained by replacing E(x) by zero, or a set of solutions of the homogeneous equation whose Wronskian determinant W(x) is not zero, where

W(x) =

For any n functions ujy W(x) = 0 if some one u is linearly dependent on the others, as un = + k2u2 + ... + hn_ûn^ with the coefficients ht

constant. And for n solutions of a linear differential equation of the nth order, if

W(x) is not zero, the solutions are linearly independent.

u2

< u2 .. 1ln' ( 6 )

Ui(n-1) 11 to"1* un


5.14* Constant coefficients. T o solve the homogeneous equation

' g + B ^ + O - O (1)

where A, B, and C are constants, find the roots of the auxiliary equation Ap2 + Bp + C = 0. If the roots are unequal quantities r and s, the solution is y = c1erx + c2esx.

When the coefficients A, B, C are real, and the roots are complex, they will occur as a conjugate pair a + bi and a — bi. In this case the real form of the solution is y = eaxc1 cos bx + c2 sin bx) or by § 2.8,

y = Qâx s m (fa Q) .

If the roots are equal and are r,r the solution is y = erx(c± + c2x). Similarly, to solve the homogeneous equation of the nth order

dy A d«~iy , A d y

where An, An_ly A0 are constants, find the roots of the auxiliary equation

Anp- + A - i ^ - 1 + :..+A1p + A0 = 0 (2)

For each distinct root r there is a term cerx in the solution. The terms of the solution are to be added together.

When r occurs twice among the n roots of the auxiliary equation, the corresponding term is erx(c1 + c2x).

When r occurs three times, the corresponding term is erx(c1 + c2x + c3x2), and so forth.

When there is a pair of conjugate complex roots a + bi and a — bi, the real form of the terms in the solution is eax(cY cos bx + c2 sin bx).

When the same pair occurs twice, the corresponding term is eax[(c1 + c2x) cos-bx + (dx + d2x) sin bx], and so forth.

Consider next the general nonhomogeneous linear differential equation of order n, with constant coefficients, or

dy d-iy dA _ E ( )

A n dx- + dx^ + ' *' + A l dx + A° " L X )

By § 5.13 we may solve this by adding any particular integral to the complementary function, or general solution of the homogeneous equation obtained by replacing E(x) by zero. The complementary function may be found from the rules just given in this section. And the particular integral may then be found by the methods of §§ 5.15-5.16.


5.15. Undetermined coefficients. In the last equation of §5.14 let the right member E(x) be a sum of terms each of which is of the type

ky kcosbx, ksinbx, keax, kx (1)

or, more generally,

hxmeax, kxmeax cos bxy or kxmeax sin bx (2)

Here m is zero or a positive integer, and a and b are any real numbers. Then the form of the particular integral / may be predicted by the following rules.

Case I. E(x) is a single term T. Let D be written for d/dx, so that the given equation is P(D)y = E(x), where

P(D) = Ajy + A^D^ + . „ + A i D + A o ( 3 )

With the term T associate the simplest polynomial Q(D) such that Q(D)T = 0. For the particular types k, etc., Q(D) will be

Z>, D2 + b2y D2 + b2

y D — a, D2 (4)

and for the general types kxmeax, etc., Q(D) will be

(D - a)m+\ (D2 - 2aD + a2 + &2)™+\ (D2 - 2aD + a2 + b2)m+1 (5)

Thus Q(D) will always be some power of a first- or second-degree factor,

Q(D)=F\ F=D-ay or F = D2 - 2aD + a2 + b2 (6)

Use § 5.14 to find the terms in the solution of P(D)y = 0, and also the terms in the solution of Q(D)P(D)y = 0. Then assume that the particular integral I is a linear combination with unknown coefficients of those terms in the solution of Q(D)P(D)y = 0 which are not in the solution of P(D)y = 0. Thus if Q(D) = Fq, and F is not a. factor of P(Z>), assume

I = (Ax*-1 + Bxq~2 + ... + L)eax (1)

when F — D — a, and

/ = (Ax*-1 + Bxq~2 + ... + L)eax cos bx

+ (Mx*-1 + Nxq~2 + ... + R)eax sin bx

when F = D2 - 2aD + a2 + b2 (8)

When F is a factor of P(D) and the highest power of F which is a divisor of P(D) is F*, try the I above multiplied by x*.


Case II. E(x) is a sum of terms. With each term in E(x) associate a polynomial Q(D) = Fq as before. Arrange in one group all terms that have the same F. The particular integral of the given equation will be the sum of solutions of equations each of which has one group on the right. For any one such equation, the form of the particular integral is given as for Case I with q the highest power of F associated with any term of the group on the right.

After the form has been found, in Case I or Case II, the unknown coefficients follow when we substitute back in the given differential equation, equating coefficients of like terms, and solve the resulting system of simultaneous equations.

5.16. Variation of parameters. Whenever a fundamental system of solutions ult w2, un for the homogeneous equation is known, a particular integral of

A ^ % + 4«-i(*> £S + - + A ^ % + A ^ x ) y = E x ) ( 1 )

may be found in the form

y = Zi vkuk (2) fc=i

Here the vk are functions of x found by integrating their derivatives vk\ and these derivatives are the solutions of the following n simultaneous equations :

JL JL

Z/vkuk = 0, 2/Vkuk=0, 2/flfc'Kfc" = 0,

(3) I

X vkWn'2) = 0, An(x) J v k ' u k ^ = E(x)

T o find the vk from vk — Jvk dx + ck, any choice of the constants will lead to a particular integral. The special choice

dx

leads to the particular integral having j , y\ y", y(n-X) each equal to zero when x = 0.


5* 17. The Cauchy-Euler "homogeneous linear equation.** This has the form

k - x n ^ + K - ^ d ^ + . . . + h x d f x + k<sy = Fx) (1)

The substitution x = e*, which makes

Xiîv x k d i = [~di~h + 1 ) • " dt~2) (dt~x)i (2)

transforms this into a linear differential equation with constant coefficients. By §§ 5.14 to 5.16 its solution may be found in the form y — g(t)> leading to y = g(ln x) as the solution of the given Cauchy-Euler equation.

5.18. Simultaneous differential equations. A system of two equations in x,y depending on t, if linear and with constant coefficients, may be written

MD)x+g2(D)y = E2(t) \ ( 1 )

where D = djdt. By §§ 5.14 to 5.16 find the solution of

[gmtm-gl(D)f2(D)]x = gjpmt)-gl(D)E2(t) (2)

in the form

x = u + c1u1 + c2u2+ . . . +cmum, (3)

and the solution of

[g*(D)UD)-glD)UD)y =fi(D)E2(t) -hD)m (4) in the form

y = v + + c2'u2 + ... + cm'um (5)

Here u, v, m 1 v um are functions of t. The constants ck and ch' are not independent, and the relations between them must be found by substitution of x and y in the original equations. In the general case, and the usual choice of u and v, these relations may be used to determine the ck' in terms of the ck.

Consider the special homogeneous linear system of n equations of the first order, with the coefficients cks constant,

~~jT = ^fciJi + ck2y2 + ... + cknyn1 k = 1, 2, . . . , n (6)

§ 5 . 1 9 BASIC MATHEMATICAL FORMULAS 3 7

( 9 )

cn\als + cn2fl2s + ••• + (cnn rs)ans — 0

5.19. First-order partial differential equations. When linear in the derivatives of the two dependent variables, the equation is

A(x9y9u)p + B(x9y,u)q = C(xyy9u) ( 1 ) where

du t du

* = Tx = u* m d ? = ^ = M* ( 2 )

The system of differential equations for the characteristic curves is dx dy du

If this is solved in the form (3)

f(xy,u) = cx and g(x,y,u) = c2 (4)

the solution of the partial differential equation is given by

F(f,g) = 0, f=G(g), or g = H(f) (5)

where JP, G, H are arbitrary functions.

5.20. Second-order partial differential equations. When linear in the second derivatives of the two dependent variables, the equation is

R(x9y)r + 2S(x,y)s + T(x,y)t = Vx9y9p,q9u) ( 1 ) where

p = ux, q = uy, r = uXX9 s = uxyy t = uyy (2

The ordinary differential equation for the characteristics is

Let the equation \cll r C12 "' Cln j I /* r v r I

= 0 ( 7 )

have n distinct roots rl9 r 2 , . . r n . Then the solution is

yk = akle^ + ak2e^ + ... + akne^ (8)

where for each s the ratios of the coefficients aks are found from

(cn — rs)îs + Ci2a2s + ••• + clnan8 = 0 c2LaU + (C22 — rs)^2s + ••• + V n s = 0


By § 5.10 its solution may be found in the form

f(*,y) = a, g(x,y) = b (4)

These are the equations of the two families of characteristic curves. The type of equation depends on the sign of the determinant

|* ST\=RT-S* ( 5 )

for any x,y region under consideration. If RT — S2 < 0, or RT < S2, the equation is hyperbolic. In this case the characteristics are real, and if the parameters a and b are used as new variables, we obtain the first normal form

uab = F(ayb,u,ua,u)) (6)

A second normal form for the hyperbolic type results from the substitution

a = X+Y, b = X-Y (7)

This is uxx wYY ~ G(x> Y,u,uXiuY) (8)

If RT — S2 = 0, or RT = S2, the equation is parabolic. In this case the two families of characteristics are real and coincident. We here use X = a = b and Y any second independent function of x,y as new variables. This gives the normal form

uxx = Y,u,ux,uY) (9)

If RT— S2> 0, or RT > S2, the equation is elliptic. Here the characteristics are not real, but use of the substitution

a = X + iY, b = X-iY (10)

leads to the real normal form

uxx + uYY = G(X,Y,u,ux,uY) (11)

The wave equation, utt — c2uxx = 0, is hyperbolic and admits the general solution

u=fx~ct)+g(x + ct) (12)

The heat equation

a2uxx = ut (13)

is parabolic and admits no general solution in terms of arbitrary functions.


Laplace's equation, uxx + uyy = 0, is elliptic and admits the general solution

u=f(x + iy)+g(x — iy). (14)

5.21. Runge-Kutta method of finding numerical solutions. For the differential equation dyjdx = f(x,y) of § 5.3 the solution may be found step by step. We start with xQ,y0, first compute xlyyly then x2,y2, and so on up to xN,yN. Here xn+1 — xn. = h is small, and at each step we find

ki = f(xn>yn)K K =f(xn + y , yn + h,

h = fxn + - j , y + ^ j j K h= f(xn + hyyn + k3)h. Then

xn+1 = xn + h, y n + 1 = y n + ky,

where A3; = 1 (k± + 2A2 + 2k3 + A4)

6. Vector Analysis

6.1. Scalars. A scalar S is a real number capable of representation by a signed coordinate on a scale.

6.2. Vectors. A vector V is a quantity which is determined by a length and a direction. The vector F may be graphically represented by any line segment having this length and this direction V = OA.

6.3. Components. Let 1, j , k represent three vectors of unit length along the three mutually perpendicular lines OX, OF, OZ, respectively, which are taken as the positive coordinate axes. Let V be a vector in space, and a, b, c the projections of Von the three lines OX, OY, OZ. Then

y=ai + bj + ck (1)

where a, b, c are the components of V, while ai, bj, ck are the component vectors. The magnitude of V is

\V\ = V= Va2 + b* + c* (2)

And the direction cosines of F a r e a/\V\, b/\V\, c[\V\ when \V\ ^ 0. When IV \ = 0, we have the null vector

0 = 0i + 0y + 0* (3)

which has length zero and no determinate direction.


i • i = 7 • j' = k • k = 1, j*y = j ' k = k * i = j ' i = k*j — i * k — 0 (5)

6.6. The vector or cross product, Fx x F2* This is defined as

Vi*V2=\V1\ I 21 sin du

where 8 is some angle not exceeding 180° from Vx to V2 and u is a unit vector perpendicular to the plane of V1 and V2 and so directed that a right-threaded screw along u will advance when turned through the angle 6. For 6 = 0 or 180°, Fx and V2 do not determine a plane, but sin 6 = 0 makes the product the null vector.

i / * • • ] K 1 x F j = - K ! x F i = a x 6! q [

«2 *2 C2 ,> C1)

= (V2 — Vi) ' + (cia2 — c2«i)/ + ( a A — a^k J x (V2 + V3) = Fi x V2 + V, x Vz (2)

( F x + F2) x K s = F 1 x F 3 + V2 x F 3 (3)

Fx x (V2 x F,) = (V, • F 3 ) F 2 - (F x • F 2 )F 3 (4)

6.4. Sums and products by scalars. These may be formed by corresponding operations on the components. Thus if

Vx = aj + bj + cxk and F2 = a2i + bj + c2k (1)

Vi+Vt= K + a2)i + (b± + b2)j + ( C l + c2)k (2)

SV= VS = (Sa)i + (Sb)j + (Sc)k (3)

(Sx + S2)V = S.V + S2V, S(VX + V2) = SV1 + SV2 (4)

6.5. The scalar or dot product, V1 • V2. This is defined as

v1-v1=\v1\ I v21 cose (i)

where 8 is any angle from Vt to F2.

V1 • V2 = F2 • Vx = ata2 + bxb2 + <y2, V1-V1=\V1 |2 (2)

Vi-(V2+r3)=rl-V2+vl-v9 (3)

( ^ + F 2 ) - F 3 = F x - F 2 + F 2 - F 3 (4)

§ 6.7 BASIC MATHEMATICAL FORMULAS _ 41

V1'(VixVa) = (V1xVt)-Va=V2-(VaxV1)=\.b1 b, bz ( 1 )

if we so number the vectors that the cross product makes an acute angle with the vector outside the parenthesis.

6.8. The derivative. Let a vector r have its components variable but functions of a parameter t, so that

r(t) = x(t)i + y(t)j + z(t)k or r = xi + yj + zk (1)

To differentiate r, we merely differentiate each component. Thus

j = = * W + y « / + = § i + %i + § * (2)

For two vectors,

rx = xxi + yj-+ zxk and r2 = * a i + j y j + z2k,

+ (4)

^ ( ' 1 x r ! ) = l X r 2 + f l x ¥ = - , x ¥ + ^ 1 (5)

p *|,|1lI ( 6 ) dt 1 1 dt v 7

Hence if | r | is constant, r - dr/dt = 0

6.9. The Frenet formulas. Let P 0 be a fixed and P a variable point on a curve in space. Take s, the arc length P 0 P , as the parameter. Then the vector from the origin O to P is

OP = r(s) = + y(s)j + z(s)k ( 1 )

(V, x V2) x F 8 = .(Fi- VZ)V2-V2- F8)Fi. (5)

i x i = 0, j x j ^ O , x A: = 0, i x j = k, j x k = 1, i (6)

k x i = j , j x i = — k x j = — i, i x k = — j )

6.7. The triple scalar product. The volume of the parallelepiped having Fx, F2, F 3 as three of its edges is

• a, a9 a* i


With each point P we associate three mutually perpendicular unit vectors t,p, and b. These satisfy the equations:

dr ^ dt 1 db 1 dp 1 . 1 . -j-=t, — = —py — = p, — = —b t (2) ds ds p * ds r ds T p

The tangent vector t has the direction of r'(s)y the principal normal vector p has the direction of r"(s), and the binormal vector b has the direction of t x p = b* The curvature lip is the length of r"(s) = dtjds. The torsion 1/T is determined from dbjds — — p/r.

6.10. Curves with parameter t. When the parameter is t,

OP = r(t) = x(t)i + y(t)j + z(t)k ' (1)

Using dots for t derivatives we form I y z \ \ z x I l x v I

r = xi + yi + and r x r = . . . * + . . . . / + . . . A : (2) 1 1 \y z\ \ z x\J \ x y\ w

Then for the unit vectors of § 6.9, the tangent t has the direction of r , the binormal b has the direction of r x r , and the principal normal p has the direction of (S^ x r ) x (S2r), where Sx and S2 are any scalar factors. The curvature and torsion may be found from

1 _ r x r 1 _ ( r x r • F )

When the parameter t is the time, the velocity vector v = vt — r , the speed

v = | v | = V * 2 + J 2 + # 2 ( 4 ) and the acceleration vector

v2

a = v = r = vt-] p P

6A I. Relative motion. A varying coordinate system with fixed origin O is instantaneously rotating about some line OL with angular velocity OJ. Let w be a vector of length co along OL. For a variable point P and i, y, k in the moving system

OP = r = a?i + 37* + # A : and r = v r e i = *i + yj + (1)

The absolute velocity of P, v a b s is given by

Vabs = Vrei + w x r = r + w x r (2)

where v r e l is the velocity relative to the moving system.


6.12. The symbolic vector del. The equation

defines the vector differential operator V , read del. The gradient of a scalar function f(xyy,z)> y / " or grad / is

g r a d / =

(1)

(2)

The gradient y / " extends in the direction in which the derivative dfjds is a maximum, and | V / 1 is equal to that maximum. In a direction making an angle 6 with the gradient, dfjds = | V / 1 cos 0.

We may apply del to a vector function of position Q,

Q(x,y,z) = Qtfayrfi + Q2(x,y,z)j + Qa(*,y,*)k

The divergence of the vector function Q, V " Q or div Q is

d i v e dx

8Qi +8 Q i

dy dz

(3)

(4)

For a fluid, let Q equal the velocity vector multiplied by the scalar density, Then div Q is the rate of flow outward per unit volume at a point.

The curl or rotation of the vector function Q, V x Q is

i 7 k

JL JL JL dx dy dz

101 02 03

curl Q = rot Q = V x Q

(5)

= / ? 0 3 _ ? 0 2 \ / , / 3 0 1 _ ? 0 3 \ ; , / ? 0 2 _ ^ 0 l \ , \ dy 0 s / \ a* dxy ^ \ dx dy)

For a fluid, let Q equal the velocity vector times the density. When the motion is analyzed into a dilatation and a rigid displacement, for the latter the angular velocity at a point or vorticity is ^ curl Q.

In terms of del the Laplacian operator is V 2 , and

v 2 / =

V • ( V x Q)

V . V / = d i v g r a d / = 0 + ^ + U ( 6 )

div curl g = 0, V x ( V / ) = curl grad / = 0 (7)

And V x Q = curl Q = 0 is a necessary and sufficient condition for Q to be the gradient of some function / ,

( 8 )


divg = v e = 1 (3)

curl curl Q = V x ( V x Q) = grad div Q~V2Q = V ( V Q)-V'VQ (9)

where

V 2 G = W e = (V*&)/ + (WzV + (V*ft)* (10)

6.13. The divergence theorem. Let S consist of one or more closed surfaces that collectively bound a Volume V. Then

jpdivQdV= jvV'QdV= $sQ-BdS (1)

where n is the unit normal to S, drawn outward.

6.14. Green's theorem in a plane. Let B be a simple closed curve in the x,y plane which bounds an area A, with the positive direction so chosen that it and the inner normal are related like OX and OY. Then

Jb (** + w*»J,(£-™)* (2, for M and N any two functions of x and y.

6.15. Stokes's theorem. Let S be a portion of a surface in space bounded by a simple closed curve B. If m is the direction into S perpendicular to J5, the positive direction for B is related to m and n, the positive normal to 5, like O Z , OF, and OZ. Then

J 5 (curie)-in*s = J s » - ( v x e ) r f 5 = J ^ e - * (1)

6.16. Curvilinear coordinates. For an orthogonal system

ds2 = hx2du2 + h2

2dv2 + hs2dzv2, dV = hxhjtsdu dv dw (1)

In terms of three unit vectors i , i2> *3 m the direction of increasing u, v, w, respectively, we have

A * I 8f . \ df . \ df . ...

§ 6 . 1 7 BASIC MATHEMATICAL FORMULAS

1

h2hz

XhQz) KKQ*)

Kk hh d. d 8

8u 8v dw KQi KQi hQs

8v dw

dw du h d(h2Q2) dih&j

du dv

Here

6.17. Cylindrical coordinates. These are r, 0, and zy where

x = r cos dy y = r sin 6

hr = 1 , ho = r, hz = 1

ds2 = dr2 + r2d02 + dz2, dV=rdr dd dz

J r dr \ dr) ^ r2 dd2 ^ dy dz2

6.18. Spherical coordinates. These are r, 6, cf>, where

x = r sin <j> cos 6, y = r sin <f> sin d, z = r cos <f> Here

hr = 1, = r , /z0 = rsin<£, ds2 = dr2 + r2d<f>2 + r2 sin 2</> d92

dV=r2 sin (f>drdc/>dd

J r2 dr \ dr) ^ r2 sin <f> dcf> \* 9 dcf>) ^r2 sin2 cf> dd2

Some authors transpose the meaning of 9 and <f> as given here.

6.19. Parabolic coordinates. These are w, v, 6, where x = uv cos 0, ^ = uv sin 0, # = 1 (w2 — a2) |

| Vx2 +y2 + z2 = \ (u2 + z;2) I

•= K =• Vu2 + a2, A0 =

<ft2 = (u2 + a 2 ) (</w2 + dv2) + u2v2d02y dV = uv(u2 + w2)<fo <fo dd

Here

J u2 + v2 u du\ du) ^ v dv \ dv) ^ \ U2 ^ V2)

dy dd2

curl Q = V x Q


7. Tensors

7.1. Tensors of the second rank. A tensor of the second rank, or second order, has nine components in each coordinate system. Let these be Apq in one system and Apq in a second system, where p = 1, 2, or 3 and q = 1, 2, or 3. If two arbitrary vectors have components Up and Vp in the first system and Up and Vv in the second system,

X UvApqV q — ^ UpApqV q (i)

is a scalar invariant, the same in all coordinate systems.

7.2. Summation convention. In equations involving tensors, like that in §7.1 the summation signs are often omitted in view of the convention that summation occurs on any index that appears twice.

7.3. Transformation of components. If, for the vector components,

the tensor components Apq transform like the product UpVq and

( 1 )

7.4. Matrix notation. We may write the nine Apq as a square array

(i) A12 A

^ 2 1 A22 A ^ 3 1 ^ 3 2 A

This is a matrix, abbreviated as || Avq || or A. Sometimes the double vertical bars are replaced by parentheses, or omitted. (See pages 85-89.)

->

A vector UP may be written as a 1 by 3 row matrix Upi or a 3 by 1 column matrix Uv\. Thus

..ui (2)

7.5. Matrix products. Let the number of columns of the matrix Apq

be the same as the number of rows of the matrix Bqr. Then the product matrix, in the order indicated, C = AB,

11 Cvr || = || Apq 11 11 Bqr 11 has Cpr — X Bgr (1)

§ 7 . 6 BASIC MATHEMATICAL FORMULAS 47

as its elements, and a number of rows the same as Apq, a number of columns the same as Bqr. Matrix multiplication is associative, (AB)C = A(BC) may be written ABC. But the multiplication is not commutative. The dimensions may not allow both AB and BA to be formed, and even if they both exist they will in general be different.

7.6. Linear vector operator. From any vector U, a tensor Avq may be used to generate a new vector by multiplication on the right,

V,= %A„U9 or Vj = \\A,a\\Uj (1) Q

Similarly by multiplication on the left

WQ = XUPA;Q or WQ=U9\\A9q\\ (2) p

7.7. Combined operators. If the first process of § 7.5 is applied with a tensor Apa, and then repeated on Vv with a tensor Bpa, the result is equivalent to a single operation with Cpq, where

Crpq = ^ BprArq or || || == || 2?pr || || ^4r ( ? || (1) r

Similarly for the second operation of §7.5, where the result is equivalent to a single operation with Dpq, where

Dpa = XA»rBra or \\Dpq\\ = \\Apr\\\\Brq\\ (2) r

7.8. Tensors from vectors. The products of components of two .vectors, or sums of such products form a tensor. Thus

Apq = BpLq+CpMq + DpNq (1)

Any tensor may be built up of not more than three such products. For Am any tensor and 2?, C, D three noncoplanar or linearly independent vectors, values of L, M, and TV can be found to satisfy the equation just written.

7.9. Dyadics. Write i2, i3 for i, j , k. Then

B = ^ Bviv, L = X Lqiq and BL = ^ BvLQiviq (1) P Q P,Q


The tensor of § 7.8 may be generated from

Âpqipiq = BL + CM + DN (2) P, Q

The first operation of § 7.7 may be effected by dot products using the right-hand factors,

V = X Vpip = X A*Mi*' U) = B(L -U) + C(M-U) + D(N- U) (3) P P,Q

Similarly the second operation of § 7.7 may be effected by dot products using the left-hand factors,

W = X Waiq = X AP«(U • ip)ia = (U-B)L + (U- C)M +(U>D)N (4)

Q P,Q

Products of vectors, or sums of such products, used as above to generate tensors and form linear vector functions are called dyadics.

7.10. Conjugate tensor. Symmetry. If Apq is a tensor, then the conjugate A'pq = Aqp is also a tensor. A tensor is symmetric if it is equal to its conjugate,

Apq = Aqv (1)

A tensor is alternating, or skew-symmetric if it is equal to the negative of its conjugate,

Apq = —Aqp (2)

Any tensor may be represented as the sum of two tensors one of which is symmetric and the other alternating by means of the identity

APQ = \ (APQ A pq) H~ "2 (APQ A pq) (1)

7.11. Unit, orthogonal, unitary. The unit matrix has

4pq = &pq = 0 if p?£q9 and 8 p q = l if p = q (2)

A square matrix is singular if its determinant is zero. Each nonsingular square matrix has a reciprocal matrix A~1

pq such that

II Avi II II A~x J l = II Kr II and || A~\t || || A„ || = (3)

A matrix is orthogonal if its conjugate and reciprocal are equal,

A' — 4 - 1 A A A A — ?\ (d\ Jix pq y x q V Q qr / / qnp-^r q u V7* \ J


For a matrix with complex elements Apq obtain its Hermitian conjugate APQ from A'vq by replacing each element a + bi by its complex conjugate a — bi. A matrix is unitary, or Hermitian orthogonal, if its Hermitian conjugate equals its reciprocal, Apq = 4~V<r

For Cartesian coordinates the matrix mvq of §7.3 is orthogonal.

7.12. Principal axes of a symmetric tensor. For any symmetric tensor Ava there are three mutually perpendicular directions such that when these are taken as a new coordinate system, the new components of the tensor have the form Apq = R^h^q. The Rv are the roots of

~~ ^ ^ 1 2 ^ 1 3

A21 A22 R A2Z

-^31 ^ 3 2 ^ 3 3 R = - R», + -S2R + S3 = 0 (1)

When this has three distinct roots, the directions of the new axes in the old coordinate system are found by solving

A21x„ + A22yp + Am?9 = RpyJf (2)

for the ratio of xpy ypy zv where p is 1, 2, or 3. The trace

&i = ^ 1 1 + ^ 2 2 + ^ 3 3 (3) .S3 is the determinant | .AVQ | and

^ 2 = ^ 2 2 ^ 3 3 "f* ^ 3 3 ^ 1 1 "f* ^ 1 1 ^ 2 2 — ^ 2 3 2 — ^ 3 1 2 ^ 1 2 2 (4)

Sv S2y S3 known as the first, second, and third scalar invariants of the symmetric tensor Avq have the same value in all coordinate systems.

7.13. Tensors in w-dimensions. In two systems of coordinates, let a point P have coordinates (x1^2,... ,xn) in the first system and (tf1,*2,...,xn) in the second system, with relations

= /*(*\#2,...,*w), x*> = Fp(x\x2y...yxn)y p = ly2y n (1)

If AxyA2

y...yAn) are related to (A1, A2y...yAn) by the equations

BASIC MATHEMATICAL FORMULAS § 7.14

by § 7.2, the ^4's are the components of a contravariant vector, or a tensor of rank one.

Similarly the relation

holds for the components of a covariant vector, or a covariant tensor of rank one.

The expressions dxm and dx1 are the components of a contravariant vector, and if ^(x1) = cj)(xm) is any scalar point function, the gradient with components df/dx™ and dfydx1 is a covariant vector.

7.14. Tensors of any rank. The scalar 0 is a tensor of rank zero. The transformation APQ = (dxv/dx*) (dxQ/dxj)Aij defines a contravariant tensor of rank two. The transformation APQ= (dxijdx'p) (dxjldxq)Ai:j

defines a covariant tensor of rank two. And the transformation A/ = (dxv\dxl) (dxjldxq)A/ defines a mixed tensor of rank two.

Tensors of higher order are denned similarly.

7.15. The fundamental tensor. This is the symmetric covariant tensor gmn such that in the Riemannian space the element of arc length is ds2 = gijdx1 dxj. In the determinant of the gij9 let Gti be the cofactor of gij9 §1.8, and let g be the value of the determinant. Then gij = G^jg is a contravariant tensor. Then gimgjm = 8/ is a mixed tensor, where § / is the

The fundamental tensors gii9gij may be used to change covariant indices to contravariant ones and conversely. For example,

7.16. Christoffel three-index symbols. These symbols (themselves not tensors) are defined in terms of the fundamental tensor by

(3)

8« of §7.11.

A*=g**Av Av=gpqA", Av" = g' (1)

(1)

r s (2) rs

They are used to form tensors by covariant differentiation, as

(3)


and dA t At _ s Pi A t I Vt A j (A\ s;r ^ r

A sr -î i A jr V /

The equations of the geodesic lines in the Riemannian space are

ds2 + 1 3,8 ds ds _ U ^

7.17. Curvature tensor. The Riemann-Christoffel curvature tensor is 5 3

This leads to Rik9 the second-rank curvature tensor, o p

# — Pr Pr i Fs pr Fs pr /9\ -tvifc ^ r

A ik i r ' r s *fc »"* fcs V~/

and to i?, the curvature scalar,

R=gikRik. (3)

All the components of each of these curvature tensors are zero for a Euclidean, or flat space.

8. Spherical Harmonics

8.1. Zonal harmonics. In spherical coordinates, for functions independent of 8, by § 7.1 Laplace's equation is

This admits as particular solutions the solid zonal harmonic functions

rnPn (cos <f>) and r^n+1)Pn (cos tf>) (2)

if Pn(cos<7J>) satisfies a certain differential equation. With cos cf> = x , and *°n — y > t n e equation takes the form

( l - x 2 ) ^ - 2 x f x + n n + l ) y = ° ( 3 )

which is Legendre's differential equation. The factor Pw(cos cf>) is called a zonal harmonic.

8.2. Legendre polynomials. For n zero or a positive integer, the only solution of Legendre's differential equation of § 8 .1, which is regular at 1 and — 1 reduces to a polynomial. This polynomial, multiplied by the factor

52 BASIC MATHEMATICAL FORMULAS § 8 . 3

which makes it 1 when x = 1 is called the nth Legendre polynomial and is denoted by Pn(x). The first few Legendre polynomials are

P0(x) = 1, Pxx) = xr P2(x) = \ (3x* - 1)

ps(x) = i ( 5 * 3 - 3 * ) > p l x ) = 8 ( 3 5 * 4 - 3 0 * 2 + 3) (1)

The general expression is

PM = 2 n - l ) 2 n T 3 ) - - - n(n-\) "2(2» — 1 )

» ( » — 1) (n — 2) (n — 3) 2 • 4(2n - 1) (2» - 3)

(2)

8.3. Rodrigues's formula

n ' _ 2"»! dx"

8.4. Particular values

P B (1) = 1, P „ ( - * ) = ( -1)"P„(*)

P 2 n + 1 ( 0 ) = 0, P 2 „(0) = (-1)< n , l - 3 - 5 . . . ( 2 » - l ) 2 - 4 - 6 . . . 2w

dx = , n „ 3 - 5 - 7 - ( 2 n + l )

, 1 ; 2 - 4 - 6 . . . 2 n

( 1 )

(1)

(2)

(3)

8.5. Trigonometric polynomials. In terms of <f> and multiples of 0, where as in § 8.1 we put cos^> = x, the first few polynomials are

P0(cos<£) = 1, P^cos^) = cos^, P2(cos(/>) == (3 cos 2</> + 1) P3(cos</>) = J (5 cos 3</> + 3 cos</>) J> (1) P4(cos 0) = (35 cos 4<f> + 20 cos 2<£ + 9)

The general expression is

P„(cos</>) 1 -3 - 5 ... ( 2 n - l ) 2nn\

cos ncf) 1 2 n - l

cos (« — 2)0

. 1 - 3 n(n — X) 1 • 2 (2w — 1) (2w — 3) v , T

(2)


8.6. Generating functions. If r < 1 and x < 1,

(1 -2rx + r*)-^ = P0(x) + rP^x) + r*P2(x) + r 8 P 3 (*) + ...

and if r > 1 and x < 1,

(l _ 2rx + r»)- i /« = - iP 0 ( x ) + ±P^x) + ±P2(x) + ...

8.7. Recursion formula and orthogonality

nPn(x) + (n - l)Pn-2(x) - (2n - l ^ W * ) = 0

(1)

(2)

(1)

(2) J^ j ^ m ( » ) ^ ( * ) dx = 0 iortn^z n, [P „ ( * ) ] 2 dx =

8.8. Laplace's integral

>(* + Vx2 — 1 COStt)n

8.9. Asymptotic expression. If e > 0 and e < (f> < TT — e,

P " ( C 0 S ^ = V ^ l n ^ 8 1 1 1 [ ( " + I K

where O means " of the order of " for large n.

+ 0 (1)

8.10. Tesseral harmonics. In spherical coordinates, by §6.18 Laplace's equation is

dr

This admits as particular solutions the solid spherical harmonic functions

rn sin md Pnm (cos </>), rn cos mO Pn

m (cos </>) J ^

r-(n+l) s m m ( 9 pm (CQS ^ a n ci r-(n+l) c o s mQ pm (CQS \

if P nm ( c o s satisfies a certain differential equation. With cos <f> — x, and

Pnm = the equation takes the form

n(n + 1) • W 2

1 3; = 0 (3)

which is the associated Legendre equation. The factor (sin md)Pn

m(cos <f>) or (cos md)Pnm(cos (f>) is called a tesseral

harmonic. When m = n, it reduces to a sectorial harmonic. And when


m = 0, that involving cos m9 reduces to Pw°(cos <j> = Pw(cos </>), a zonal harmonic.

Solid harmonics are often combined in series or integrals which for fixed r and </> reduce to Fourier series or integrals of known functions of 6. (See §§ 14.10 to 14.16.)

8.11. Legendre's associated functions. Let m and n be each zero or a positive integer with n^m. Then there is only one solution of the equation of §8.10 which does not have logarithmic singularities at 1 or —1. With proper scale factor this solution is called the associated Legendre function and is denoted by Pn

m(x).

Amp (x\ 1 Jn+m P^x) = (1 - *»)»/« - ^ 1 = (1 - * 2 ) r o / 2 (*2 -

p m(v\ (2n)\ y2\m/2 fn W - 2"n$n-m)l^~X

dxn+m v

(n — m) (n — m — 1) 0 , 2 ( 2 * - 1 )

(n — m) (n — m — 1) (« — m — 2) (n — m — 3) •2-4(2» —l) (2n —3)

8.12. Particular values

Pn\x) = Pn(x), P™-X) = -$nPnmx)

pn(x) = (Ml (l _ *»)»/*= 1 • 3 • 5 ... (2n - 1) (1 - x2)"/2

With x — cos </>, the sectorial harmonic

Pnn(cos(f>) = 1 • 3 • 5 • ... (2« — 1) sinM<£

(1)

(2)

(1)

(2)

8.13. Recursion formulas

(n - m)Pn™(x) + (n + m- l)Pn-2m(x) - (2n - l)xPn^(x) = 0 (1)

Pnm+2(x) + (n-m)(n + m+ l)Pn

m(x) )

x 2(m + 1) Vl —x2

Pnm+1(x) = 0

8.14. Asymptotic expression. If e > 0 and e < c f ) < r r — e,

P„™(cos</>)

(" + T)* + T: + TflTT

~2 + 0 ( / Z w ~ 3 / 2 )

riTT sin "

where O means " of the order of " for n large compared with m.

(2)

(1)


8.15. Addition theorem. Let y be defined by

cos y = cos</> cos<£' + sin^ sin</5' cos (0 — 6') (1)

Thus y is the distance on the unit sphere between the points with spherical polar coordinates (1, #,(/>) and (1,0',<£'). Then

Pw(cos y) = Pn(cos(/>)Pn(cos(l)')

8.16. Orthogonality

J"^ Pnm(x) Pk

m(x) dx = 0 iork^tn,

(1)

. J -l n w In + 1 (n — m)!

9. Bessel Functions

9.1. Cylindrical harmonics. In cylindrical coordinates, by § 6.17 Laplace's equation is

This admits as particular solutions the harmonic functions

e~az s'mnd JJar^, e~az cos n8 Jn(ar) ) ] (2)

eaz sin nO Jr/ar), eaz cos n6 Jn(ar) \

if Jn(ar) satisfies a certain differential equation. With ar — x and / w = yy

the equation takes the form

X * ^ + Xdx + x 2 - n 2 ) y = ° ( 3 )

which is BessePs differential equation.

9.2. Bessel functions of the first kind. Only one solution of the equation (3) of § 9.1 is finite for x = 0. With proper scale factor it is called the Bessel function of the first kind and is denoted by Jn(x)9 where n ^ 0. For n zero or a positive integer


In particular, T / \ -f t'V ( i/V uv uv // X

VoW = — 22" +• 2 4 (2 ! )2 ~ 2 6 (3! ) 2 + 2 8 (4!) 2 ~"~ "*' ^ '

/ \ tA/ i/V UV UV UV / \

i W = y — 2 ^ + 2 ^ 2 ! ^ ~ 2^3!4! + 2MW.~'''

The expression involving the gamma function of § 13.1.

/*(*) = 2/ 2P+a*Jfe! r ( p + & + 1) ( 4 )

reduces to the above when p = n, zero or a positive integer, and for other positive p defines the Bessel function of order p.

9.3. Bessel functions of the second kind. When p is positive and not an integer

°° x ~ p + 2 k

/-„(*) = 2/ 2 - * + * * A ! r ( - ^ + ^ + T y ( 1 )

is a second solution of Bessel's equation of order p. Or we may use

YP(x) = NP(x) = ^±j- [cotpnUx) - / - , ( * ) ] (2)

For n = 0 or a positive integer, J-Jx) = (—1 but as -> w, the limiting value of ^ ( f f ) is YJx)> where

Ynx) = \ 77

2Jn(x) In y - g 2 ^ 4 i ( i » + J f e ) l ^ + W ) + ^ ) ]

\ (3) ^ ( » _ r - l ) ! ^ + 2 » "

2~n+2rr\ J

ijj(k) = - 0.5772157 + 1 + 1 + 1 + . . . + 1 (4)

9.4. Hankel functions. The Hankel functions, sometimes called Bessel functions of the third kind, are defined by

= JM + iYJx) = s " i ~ i^JM ~ / - * ( * ) ] (1)

H,m(x) = - iYJx) = - ^ [e^JP(x) - J-P(x)] (2)


9.5. Bessel's differential equation. Let Zv(x) be a solution of

For p nonintegral,

Zp(x) = cjp(x) + cj^(x) (2)

and for any positive or zero value of p, ZP(X) = C1JP(X) + C2YJ,(X)- (3)

or Z9(x) = cxH™x) + cM™x) (4)

give general solutions with cly c2 arbitrary constants.

9.6. Equation reducible to Bessel's. The equation

X*^+l(l-2A)x-2BCXc+1]d£ )

+ [(A* -By) + BC(2A - C)xc + B2C2x2C + E2D2x2E]y = 0 (1)

In terms of the Hankel functions

Ux) = ±[H™(x) + H™(x)], YP(x) = ±-.[H^(x)-H^(x)] (3)

Let A be a path in the complex plane made up of a small circuit of the point 1 in the positive direction and two lines parallel to the imaginary axis in the upper half plane extending to infinity. Thus A starts at infinity with an angle —37r/2, and ends at infinity with an angle TT/2 (see § 20.3.) Then for z with positive real part

Similarly for B a path made up of a small negative circuit of the point —1 and two lines parallel to the imaginary axis in the upper half plane to infinity, starting with an angle TT\2 ancf ending with an angle —37r/2,

#p ( W (*) = ^ # jBeiz>2 - ! ) P -112 d u ( 5 )

The Hankel functions are complex for real values of x. But i*+*HMiy) and i-^H™-4y) (6)

are both real when y is real and positive. When the imaginary part of z goes to plus infinity, Hp

(1)(z) approaches zero. And when the imaginary part of z goes to minus infinity, Hp

(2)(z) approaches zero.


becomes Bessel's equation of order p, in Z and X if

y = xAeBxCZ, X = DxE

and so has as its solution y — xAeBxCZv(DxE).

And in particular y = xAZvDxE) is the solution of

x* + 0 - 2 A ) X T~ + K a ~ E*P*) + wtPx^y = 0

The solution of d2y\dx2 + Zfoy = 0 is

y = ViZ,/3^jVBofll^j

9.7. Asymptotic expressions. For x large compared with p,

H™(z) = i-'-WlUtiirzyVW Sp(-2iz)

H™z) = iv+1l221l2(7Tz)-V2e-iz SP(2iz)

where the asymptotic series for Sp is given by

o r r t - i • | ( 4 j > 2 - - l ) ( 4 j > 2 - 9 ) • ^ W - • n 4 f I - " 2 ! ( 4 0 2

( 4 ^ - l ) ( 4 j > 2 - 9 ) ( 4 / > 2 - 2 5 ) ^ 3! (4*)» ^ " '

If

<f>P = « — T a n d S„(2ix) = P„(x) — iQv(x)

with the series Pj, and <3„ real, JP(x) = 2V%TTX)-V\PP cos0s - 0 , sin<^)

Yv(x) = 2 1 / 2 (TTX)-I / 2 (P j , sin <f>v + 0 , cos

Using the leading terms only, and O meaning " of the order of,"

JJx) = 2V\nx)-^ c o s 4 , + o(±i^

F ^ ) = 2 ^ M - 1 / 2 s i n ^ + O ^ j

9,8. Order half an odd integer. Let p have the form

n an integer or zero.

§ 9.9 BASIC MATHEMATICAL FORMULAS

The expressions of § 9.7 assume a closed form. The first few are

/1/2M = 21I\TTX)-^ sin x

/ _ „ , ( * ) = - F_ 1 / 2 ( * ) = T-I^TTX)-^ cos x

J3,2(x) = 2 ^ M " 1 / 2 ( - cos x +

J-3l2(x) = Ym(x) = 2 ^ x ) - ^ - sin x -

Y]>(x) = (-l)^J_v(x) if p = n + ±-

9.9. Integral representation

1 ~ ~~ J o c o s c o s M ~ nu)du

9.10. Recursion formula

+ = ^ Jvx), Y^(x) + F , + 1 ( * ) = I F ^ )

9.11. Derivatives

^ _ _ l r , r - A . T - T - - T dx x 2 , - 1 # 2 > + 1 2 3 5 - 1 2 3 3 + 1

^ [**/,(«*)] - ax*]^(ax\ [xr*Max)] = - ax^Jâx)

and Yv(x) satisfies similar relations.

9.12. Generating function

e(x/2)(t-llt) = J ^ t n

n=— 00

9.13. Indefinite integrals

r r/ \ T / L u bxJJax)Jp-1(bx)—axJv-1(ax)J!P(bx) \xjip(ax)]v(bx)dx = JVK u v 1 V J _ b 2

x2

jx[Jv(ax)]2 dx = — [Jp(ax)f - ]âx)]âx)

9.14. Modified Bessel functions. We define Ivx) = i~p Jp(ix), where i~p = e-*^/ 2 = cos y — i sin y


For p not zero or an integer we define

K*(X) = 2^p-nV-»(-X)-1^ W

For p — n, zero, or an integer,

Kn(x) = ( -1) -»J„(*) ^ y + y X ^ oZl!^ r=0

> (5) 1 ^ rn+2k

+ ( - i r T X 2 ^ 1 ( w + , ) ! [ ^ + »)-^)] ^

where 0(6) is that of (4) of § 9.3. For all values of p,

The general solution of the differential equation

1 S y = cih(x) + ^ 3 > ( * ) ( 7 ) and the general solution of the differential equation

l s > , = * i / » ( * ) + * 2 ^ ( * ) + ^ ^ w

10. The Hypergeometric Function

10.1. The hypergeometric equation. The differential equation

x ( i - * ) S + [ c _ ( f l + * + i ) * ] ! - ^ = o ( 1 )

has one solution regular at the origin. With a scale factor making it 1 when x = 0, this solution is called the hypergeometric function and is denoted by F(a,b \c ;x).

10.2. The hypergeometric series. For \x \ < 1, F(a,b\c;x) equals a • b a(a + \)b(b -+ 1) ,

1 + 1 • c + 1 • 2 • c(c + 1) X

(1) a ( a + l ) ( a + 2)l>(Z>+l)(Z> + 2)

1 - 2 - 3 l ) ( c + 2) * - J


= (1 - x)-*F(b,c - a; c; ^ - j j

(2)

is one set of four. As six distinct ones we may take

F(ayb \c ;x) x^la + 1 — cy b + 1 — c; 2 — c; x)

F(a, b\a + b+\—c\\—x) (1 — xf-a-bF(c — ayc — byc+l—a — b;l—x)

1 x~aFây a+l—c;a+l—b-y

x~bF(by b + l - c ; b + l - a ; - p j

F ( a ' b ; c ; X ) = rtc-^Tic-f) F(a,b-ya+b+l-c;l-x)

+ (1 _ xy-a-b F ( g ) p g + * " C)Fc - ay c - b ; c + 1 - * - b ; 1 - *) j

(3)

(4)

; g ;*) = gjggg g -x)-^(ay a + 1 - c ; a + 1 - 6 ; 1 ) (5)

With T(c) as in § 13.1,

F(ayb;c;x) __ a(a + 1) ... a + n) (b + 1) ... (b + n) \ £2. T(c) ~ (n+l ) I ( 2 )

F(<z + w + 1,6 + n+ l;n + 2;x) )

10.3. Contiguous functions. In general, a second solution of the differential equation of § 10.1 is given by

x1~cF(a+ l - c y b + l-c;2-c;x) (1)

There are other functions related to F(ayb \c \x) making in all a set of 24, and a number of linear relations connecting them. They form six groups of four, each set of four being equal except perhaps for sign.

For example, F(ayb \c \x) = (1 — x)c~a-bF(c — a,c — b;c;x)


10.4. Elementary functions (l + x)" = F(-n,b;b;-x) (1)

l n ( l + *) = * F ( l , l ; 2 ; - * ) (2)

« * = Hm' F(l9b;l',xlb) (3)

cos nx = F(^n9 —^n ; \; sin2 x) (4)

/ • vp /1 + . » 1 —•» 3 . o \ Vex sin w# = w(sin x)F I —-—, — - — ; — ; sin'2 x\ (p)

(6)

s in- 1 x = xF\-^,^-;^r-;xA (7) \2' 2 ' 2 ' J

tan-1 * = xF^Y' 1 '• y ' — * 2 ) (8)

10.5. Other functions. For polynomials, see § 10.7. For the Pnm(x)

of §8.11 : p m ( x ) > + l ) m / 2 1 J ^

* W = j o ' V l - A 2 s i n ^ ^ = 1,1; 1 -Ik* j

(1)

(2)

(3)

10.6. Special relations

-^F(a,b ;C;x) = ^-F(a+l,b+l;c+l; x) (1)

1 ' ' ' l ) T(c-a)T(c-b) K )

F(a,b ;c ;x) = r ( f t ) r fr_ft) J ^ ( l ~ " " ( l ~ d u (3)

F(— w + »» + 1, 2m + 2 + k; 2m + 2 ; *) \

(2m + k+ 1 ) ! * 2 m + 1 rfx*1


10.7. Jacobi polynomials or hypergeometric polynomials

Jn(P>4 ">*) = F(~n> P + n;q\x) \

lA-Qf] x \ q ~ p dn 0-) ~~q(q+l)..-(q+n-l)dx"lX ( 1 *' 1 )

For < ? > 0 , p > q — 1, these form a set of polynomials orthogonal with a weight function w(x) = a^-^l — fl)*-* as in § 14.19 on 0,1. They satisfy

4l~x)^+[<l-(P+m%+(*)= l (3)

and the Legendre polynomials of §8.2 are orthogonal on —1,1. For

Tn(x) = cos (» c o s - 1 x) = Jn (0, y ; ] ]

and the Tschebycheff poylnomials Tn(#) are orthogonal with weight function IIVI - * 2 on - 1 , 1.

10.8. Generalized hypergeometric functions. Let (a)0 = 1, (a)n = a(a + 1) (a + 2) ... (a + * — 1). And define

pFq(ai, aK •••> ^2>5 •••> kq\ X)

= V ( f l l ) A ) n - f e ) n * \ , 3 (*!>»(*•)» •..(*«)« » ! " f (1)

= i I a i a z - a * x , a i ( a i + iWâ + 1) + 1) x*_ 'rb1b2...bQ ^ bl(b1+ l ) f t 2 ( f t 2 + l ) . . . bq(bq+l) 2 ! ^ - /

For example, e» = ^ ( f t ; b ; *) • ^ ( a , ft ; c ; *) = ft ; c ; *) (2)

of §10.1.

The Bessel function of § 9.2,

h(x) = i ^ i ( n + - j ; 2» + 1; 2ix) (3)


= (l - *)-V.(-f-' a + 1 ~ b ~ c> (4)

a + l _ M + l _ c ; _ _ ^ _ ) )

3F%(a, b, c; a + 1 — b, a + 1 — c; 1) \

r(a /2 + 1 ) I > + 1 - b)V(a + 1 - c ) I > / 2 + 1 - ft - c ) ( 5 ) T(« + l ) r (o /2 + 1 - b)Tajl + 1 - e)T(a +\-b-c) J

10.9. The confluent hypergeometric function. M(a,c,x) = JFâ ;c ;x) is a solution of

11. Laguerre Functions

11.1. Laguerre polynomials. The differential equation

has polynomial solutions known as Laguerre polynomials and denoted by LJx). The first few are

L0(x) = 1, Lxx) = —x + 1, L2(x) = x2 — 4x+ 2 ) (2)

L3(^) = —x*+ 9x2- 18* + 6 S In general

Ln(X) = - 5l + ~ I-)2 + _ + ( _ l ) n M , j ( 3 )

^ ( * ) = e * ^ ^ = ^ ( - » ; i ; * ) ( 4 )

11.2. Generating function p-xtl(X-t) ™ fn


L n + 1 ( * ) _ (2n + 1 - *)£. (*) + n2Ln^(x) = 0 (1)

aF2(a, b,c;a+l—b,a+l—c;x) ,


11.4. Laguerre functions. The Laguerre functions e~xl2Ln(x) satisfy the differential equation

* 5 + £ + ( T - T + - W <•» These functions are orthogonal on the range 0, » , and

J o e-*Ln(x)LJx)dx = 0 if m^n; j q e - * [L M (* ) ]V* = (nlf (2)

11.5. Associated Laguerre polynomials. These are the derivatives

L * x ) = !*• [ L ' x ) ] = ^ ( t + t ) 1 * l F l i ~ r ' s + 1 ' x ^ M They satisfy the differential equation

*S+('+1-*)J+('-')y = 0 (2) 11.6. Generating function

11.7. Associated Laguerre functions. In §11.5 put r — n + k and s = 2k + 1. Then the associated Laguerre functions are defined as

er*l*x*Ln+i*»-\x) (1)

For x> the r of spherical coordinates, the volume element involves x2 dx, and with this element the functions, each times ar~1/2, are orthogonal.

e-xxkLn\x)Lm\x)dx = 0 if (2)

e~*x«[Ln\x)]2dx = (3)

12. Hermite Functions

12.1. Hermite polynomials. The differential equation

g - ^ + ^ . O (1,

has polynomial solutions known as Hermite polynomials and denoted by Hn(x).

66 BASIC MATHEMATICAL FORMULAS § 1 2 . 2

The first few are H0(x) = 1, H^x) = 2xy H2(x) = 4x*-2

H3(x) = 8*3 - 12*, H^x) = 16*4 - 48* 2 + 12 In general, with K equal to the biggest integer in w/2,

K

(2)

#»(*) = X ( -1) S : T ! « ( » - ! ) •••(».-2^ + l)(2«)»-» (3)

/ / n W = ( - l ) V 8 ^ (4)

H w = ( - l ) "2" (2n - 1) (2n - 3) ... 3 • 1 ^ ( - n ; 1 ; * » ) (5)

^ 2 n + i = (-1)"2»+H2n + 1) (2» - 1 ) . . . 3 • 1 v^ i ( - » ; j ; * a ) (6)

12.2. Generating function

e-t>+2tx = e*v«-*>2 = X (1)


Hn+i(*) ~ 2xHn(x) + 2nHn^(x) = 0 (1)

12.4. Hermite functions. The Hermite functions e-x2,2Hn(x) satisfy the differential equation

g + ( 2 n + l-x*fr = 0 (1)

These functions are orthogonal on the range — °o, <», and

H^)%^^) r - " ' ^ * ) ^ * ) ^ = 0 if rn^w j (2)

e - * 2 [ # w ( * ) ] 2 = 2 n » ! )

13. Miscellaneous Functions

13.1. The gamma function. For p positive the integral

V(p) = xê~xdx (1)

defines the gamma function.


The infinite products

J = en*-n f f ( l • + zJzl\e-(*-»ln = z e v z fx 11 + l-\e~zjn (2)

r(*) L\ \ n ) Ll \ nJ

define the function for all complex values of z.

y = 0.577216 (3)

n/ x r ( 1 - 2 - 3 ... n)nz /yJX

r(«)= lim - j ^ r-^r f - . r~rr (4) v J n^co z(z + 1) (z + 2) ... + n + 1)

13.2. Functional equations

r ( * + i ) = *r(«), r ( « ) r ( i - ^ ) = s m ^ (i)

r(~-jr(?-^j... r ( ^ + ^ ~ 1 ) M Z " 1 / 2 = (27r)<«-i»/2r^) (2)

r ( | ) r ( i ± i ) 2 * - i = v ; i > ) (3) 13.3. Special values

T ( l ) = 0! = 1, I » = (n - 1)1 = 1 • 2 • 3 ... (n - 1) (1)

for « a positive integer. r ( l ) = ^ (2)

13.4. Logarithmic derivative

a)

If the terms of a convergent series are rational functions of n> by a partial fraction decomposition the series may be summed in terms of ifj and its derivatives by the series of this section.

13.5. Asymptotic expressions. If O means " of the order of,"

In I » = In [ V 2 S « - - ^ + ^ + O ( l ) (1)


x\ = F(x + 1) = Vzirx'+ê-1 + l i + ° ( ^ ) (2)

The first term of this, A/277 nn + l / 2 e~n, is Stirling's formula for factorial n, and for large n may be used to evaluate ratios involving factorials.

13.6. Stirling's formula Til

Hm ~ 7 ^ = 1 1 / 0 = 1 (1) nâo V277 Tln +1/2 e~n V '

13.7. The beta function. For^> and q positive

B(P>4) = jl xP-\l - x)q-x dx = 2 ^ 2 cos 2 *" 1 8 s in 2 *- 1 6 dO (1)

In terms of the gamma function

m p + l . D - ^ B i p , , ) , (3)

13.8. Integrals. In the following integrals, the constants are such as to make all arguments of gamma functions positive.

/ : * - ( ' 4 ) ' ^ < ? T T F T j;~^-i.~«.r('-±!) (.) The double integral of xpy® over the first quadrant of xja)A + (y/b)B= 1

is

r (^ i + £+i + . The triple integral of xpyQzR over the first octant of

(3)

§ 13.9 BASIC MATHEMATICAL FORMULAS

Jo 2« J o 2 « 2

13.10. The Riemann zeta function. For z = x + zjy, # > 1,

1 «*) = 2 £ = TT ( i - r - ) - 1

n — p i i m e where the product extends over all prime integers p.

For x > 1,

«*) = r w J o ? I R R *

For x > 0,

t w " ( i - 2 1 - * ) i » J o f * + i f l r

14. Series

14.1. Bernoulli numbers. These are generated by

x „ xk

i ? 7 = 0, 2?8 = — 3 ~ 0 ' ^ 9 = 0, B1Q = ~6%, i ? n — 0, # 1 2 — -2~7~~

13.9. The error integral

Erf* = - ~ [*<r*dt VTT JO

The factor outside makes Erf ( a>) = 1. For small x,

' , 2 1 x* , x* \ E r f ^ ^ ^ - m + ^ - . . . j

For large x the asymptotic expression is

t L r t x - i V T T ^ \ 2x* + ( 2* 2 ) 2 ( 2* 2 ) 3 + " 7


14.2. Positive powers. For m any positive integer, the finite sum (B 4 - n)m+1 — Bm+1

\m + 2™ + 3m _|_ + n m = z K° ~T n) , ] y

m + 1 where on the right is meant the result of expanding by the binomial theorem § 1.3 and then replacing powers Bk by the Bernoulli numbers Bkoi § 14.1. In particular

l + 2 + 3 + . . . . + » = J « ( » + l ) (2)

p + 2 2 + 3 2 + ... + n2 = ± n n + 1 ( 2 w + 1 ( 3 )

l 3 + 2 3 + 3 3 + ... + rc3 = I w2(w + l ) 2 (4)

14.3. Negative powers. The sum of reciprocal even powers

i + _ L + _ L , _ L + .... (-ir(27r)^B2m_ 1 ' 22m ' 32m "i" 42m -r ••• — 2(2m)\ ~~ ^

In particular,

This shows that for large m, the numerical value B2m becomes infinite like

2(2m)\ (2tt)«

which exceeds (7w/10)m.

(3)

14.4. Euler-Maclaurin sum formula. For m and n positive integers,

%m=\mjm+y w o ) + A m )

(i)

+ % (§y![/<2fc"1,(M) - / ( * * ~ 1 , ( 0 ) ] + / ( h , + 2 , ( 0 » ) where 69 is a suitable number in the interval 0 < 8 < 1.

14.5. Power series. For simple functions power series may be obtained by use of Taylor's theorem of § 3.25. For the series listed, the expression following a series indicates a region in which the series converges. No expression is added if the series converges for all values of x.


(n - &)!&!'

** , * 3

(i) * i < i

!„(! + *) = * * + * * + . . . , | x | < i ( 3 )

^3 OC^

sm x = x—-^ + -^ — -^+ ... (4)

COS* = 1 — + -rr~ -yr + . . . (5) *2 *4 *6

6 T + ' 2! + 4! *6

6 T + '

*3 2* 5 17*7 + 62X9 + ... ^ 2 8 3 5 ^ T + 15 " f 315 + 62X9 + ... ^ 2 8 3 5 ^

*2

2f +

5*4 61* 6

" 4 T + "6T , 1385*8 ,

+ 8! +

*3 1 -3 1 - 3 - 5 x1

T + 2 - 4 2 - 4 - 6 7

*3

3~ +

X5

5 ~ ~ *7

| x | < 1

*3 2* 5 2 - 4*7 j 3 • 5 3 •: 5 -7 1

2* 3 , 2 - 4*5 2 - 4 - 6 * 7

" 3 + 3 •5 + 3 - 5 - 7 +

*3 *5 *7

3! + ' 5 ! + " 7 ! +

V. 4! ^ 6! ^

Xs 2* 5 17*7 62* 9

• y + 15 315 1 2835

s e c * = 1 + — + ^ r - + —oi h | * | ( 7 )

sin" 1 * = * + ^ + ^ - 4 ' ^ + nmJ

Amz'A;+ | * | < 1 (8)

(9)

Vl — ^ 2 sin- 1 * = * — — T7T~" . T. i ~ •> I * I < 1 ( 1 0)

VI—* 2 = * ~ r ~ ~ r ^ o . ^ . t " T - - - » | * | < 1 (11)

s inh* = * + ^T+^T+ ... (12)

(13)

77

tanh* = * - — + - r ^ - - ^ + | * l < - 2 ( 1 4 >

14.6. Elementary functions

, n(n — l) 0 , n(n — l)(n — 2) «, , (1 + ocf = 1 + nx + v ; x2 + -± 3 7 ) *3 + ..

BASIC MATHEMATICAL FORMULAS § 1 4 . 7

s inh - 1 * = x — L x* 1 ' 3 *5 1 , 3 , 5

2 ' 3 + 2 • 4 ' 5 2 - 4 - 6 * 7

1*1 < 1

s inh - 1 * = In 2x + 2 ' 2* 2 2 • 4 ' 4* 4 + 2 • 4 • 6 ' 6*6

I * I > 1

c o s h - 1 * = In 2* ~ 1 - 3 - 5 1

2 2* 2 2 - 4 4*4 2 - 4 - 6 6*6

* I > 1

t a n h - 1 * = * + y + y + y + 1*1 < 1

-yi3 >yi5 <yi7 r> _ IA* t/V L\

^ & = , _ _ + _ _ _ _

14.7. Integrals

r .

cos (x2)dx

sin (*2)d*

*° ** *• ,13

5 - 2 ! 1 9 - 4 !

v7 *' *x

13 • 61

«15 * A '

7 - 3 !

r* sin * Jo * dx = x

1 1 - 5 ! 1 5 - 7 !

*5

foo COS t

3 - 3 ! 1 5 - 5 !

= 0.577216 + In * 2 - 2 ! ' 4 - 4 ! * > 0

(15)

(16)

(17)

(18)

(1)

(2)

(3)

(4)

(5)

f J X

dt = -0 .577216 - l n * + * -2 - 2 ! ' 3 - 3 !

., * > 0 (6)

14.8. Expansions in rational fractions. In this section the prime on a summation means that the term for n — 0 is to be omitted

cot z 1 °° /

4 + 2 1

z — rvn tiiT \7T z z2 — n2

CSC Z 1 00 ,

Z — 7VTT tlTT ITT Z f"* J 71=1

( - 1 ) " 2z

z2 — H2TT2

( 1 )

(2)

BASIC MATHEMATICAL FORMULAS 73

csc2z = y, 1—r^, s e c * = S, ( — l ) * " 1 ^ 4 ^ f , 2 o^ 7 7 , 2 (3)

14.9. Infinite products for the sine and cosine. The prime means omit n = 0.

oo / \ oo / 2 \

n = — o o \ / n=l \ I

00 *

cos # = ]~[ 4*2

(2n - 1)2772 (2)

14.10. Fourier's theorem"for periodic functions. The function f(x) is periodic of period p if

/ ( * + * > ) = / ( * ) (i)

The period p and frequence o> are related by the equations

co = — and p = — (2)

Then for any constant c, the Fourier coefficients are

* * ' (3)

bn = JC+/>/(^) sin wcox

It is often convenient to take c = 0, or c = —p\2.

The Fourier series for f(x) is

00

f(x) = a + ^ ( « n cos rcco# + 6 n sin wcotf) (4) n=l

A regular arc is a continuous curve with finite arc length. The function f(x) is piecewise regular if its graph on any finite interval is made up of a finite number of pieces, each of which is a regular arc or an isolated point. For any piecewise regular periodic function the Fourier series will converge to f(x) at all points of continuity, and to \[fxJr) + / ( #—) ] at all points of discontinuity. H e r e / ( * + ) is the limit approached from the right, and/(#—) is the limit approached from the left, at x.

BASIC MATHEMATICAL FORMULAS §14.11

14.11. Fourier series on an interval. For any function / ( * ) , defined for c < * < c + p, but not necessarily periodic, the relations of § 14.10 may be used to find a Fourier series of period p which represents f(x) on the interval c, c + p.

14.12. Half-range Fourier series. For any function / ( * ) , piecewise regular on 0,L the Fourier cosine series of period 2L which represents / ( * ) for 0 < x < L is

where

00

f(x) = a + ^ an c o s n a ) X

n=l

a• = - i - f L / ( * ) ^ * , n = y - f L / ( * ) cos wo>* </*, co = -y-

The Fourier sine series of period 2L for / ( * ) is oo

/ ( * ) = ^ &w sin wou* n=l

2 where

bn = — j Qf(x) sin ^> 6 0 = Z~

14.13. Particular Fourier series. For 0 < x < L, 4 / . 7TX , 1 . 3 7 7

1 = * ( 8 M R + Y 8 M T >

377*

1 . 577* T s m — ,

2L I . 77* 1 . 277* 2 ~ S m ~L~" T S M X

A* + 5 = —

If

4L " 772

( 4 5 -

77* , 1 377* , 1 577* c o s — + 02 C 0 S T ~ + *2 C 0 S "T~ L

2LA) sin

3 2

77*

L

2LA

5 2

277*

L

4B + 2LA

2 S m " L

377* 2LA . 4T7* , T 8 m T + " '

and

/ ( * ) = JF/ for c < * < c + w, / ( * ) = 0 f o r c + & ; < * < c + 2L, / ( * + 2L) = / ( * ) ;

then for this periodic square pulse,

Hw 2H 2L

fx). X0 0 1 . nrrw nn — sm — — cos — | * «_i n 2L L

a) (2)

(3)

(4)

( 1 )

(2)

(3)

(4)

(5)

(6)

§ 1 4 . 1 3 BASIC MATHEMATICAL FORMULAS 75

For 0 < * < 77,

77

T 77

T

77* 77Z

** T

77* 77* 2 4 :

77*" 77** — ^- —

12 8 24

In cos —

In tan — 'o 2

In I 2 sin w

du

sin 3* , sin 5* , sin 7* sin * +

sin 2x | sin 4* sin 6* ' I i i z i • • • 2

cos * +

cos 2* ~ 2 2 ~ ~ "

4 1

cos 3* 3 2 +

cos 4*

6

cos 5* cos 7* 52 •

cos 6* 42

, sin 3x , sin 5* , sin 7* , sm ^ H _ + — ^ - H = = — +

sin 2* sin 4* sin 6* ~ 2 3 _ + " 4 s - H S3"" +

In sin — = — In 2

= - In 2 +

cos * cos 2* cos 3*

cos * cos 2* , cos 3* 1

~ / sin a

- 2 T

sin 3* , sin 5* ~ 5 2 _ ~ 32 ' + ^2 ••)

sin 2* sin 4x , sin 6x

tan -

2r sin x 1 - r 2

2 2 1 4 2 1 6 2 1 " '

X 2n — 1 s i n ^ 2 n ~~ ' r ' < 1

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(15)

(17)

1 , 2r cos * / x -. r"7"-" _ _ , ,

y t a n " f T T T 2 " = X 2^1 c o s 2 n ~ l ) x ' I r 1 < 1

(18)

For < x < 77,

2 sinh #77 + 2 ( - i r a cos >z* — w sin nx (19)


cos ax = 2a sin arr

sinh ax =

cosh ax =

2 sinh an

2a sinh arr

1 cos x cos 2* cos 3* 2a1 + a2" ~ iF^ti* 3 2 - a 2

s in* 2 sin 2* 3 sin 3* l2 + a2 2 2 + a2 1 3 2 + a2

cos * cos 2* cos 3* 2 « 2 l 2 a2 3 2 + a2

(21)

(22)

(23)

14.14. Complex Fourier series. The Fourier series of § 14.10 may be written in complex form as

uu

where

P C + P f(x)e-inoiX dx, to :

J c

2m P

T h e complex Cn are related to the coefficients of § 14.10 by

•n l VKJn

(1 — r cos co*) + ir sin co* 1 + 2r cos cox + r 2 XrUe r I < 1

In 1

Vl — 2r cos co* + r 2 i tan~ r sm 6o* 1 — r COS 60* T 72

71=1

^ 1 . )

(i)

(2)

(3)

(4)

(5)

r I < 1

14.15. The Fourier integral theorem. Let / ( * ) be piecewise regular, so redefined at points of discontinuity that

/(*) = *[/(*+)+/(*-)]

and such that J* | / ( * ) | */* is finite. Then

1 roo. Too

/(*) = — J ^ J cos u(x — t)f(t) dt

In complex form,

/ ( * ) = lim 4- [A du f°° eiu^f(t) dt

( l )

(2)

(3)

§14 .16 BASIC MATHEMATICAL FORMULAS 77

14.16. Fourier transforms. The Fourier transform of f(x) is

F(u) = ê-iutf(t)dt (1)

and by § 14.15,

/ W = A™ i * W d u & In a linear system, if R(x,u) is the response to eiux, the response to f(x) is

* w = ife 2I / I rftt (3) 14.17. Laplace transforms. If/(£) =• 0 for / < 0, its Laplace transform

is F(p) = ê-vtftfdt (1)

We write F(p) = Lap / (*) . For derivatives,

Lap / ' ( / ) = -/(0+) + p Lap A*) (2) Lap / " (0 = -/'(0+) -PA0+) + />2 Lap /(0 (3)

and so on. These relations reduce a linear differential equation with constant coefficients to an algebraic equation in the transform. We have

tn 1 k L a p ^ = ^ T ' ^ P £ - a t s i n k t = p + ay + kz ( 5 )

I f < - " < * » = & + L a P 1 = 7 ^

h(t)= \'f(u)g(t-u)du, ) (7)

LapA(i )=[Lap/ (<) ] [Lap^(0] )

For examples of the use of Laplace transforms to solve systems of differential equations, see Franklin, P., Fourier Methods, McGraw-Hill Book Company, Inc., New York, 1949, Chap. 5.

14.18. Poisson's formula. For small x> the right member converges rapidly and may be used to compute the value of the left member in


14.19. Orthogonal functions. Many eigenvalue problems arising from differential systems or integral equations have eigenfunctions which form a complete orthogonal set. For these functions <f>n(x)

$<f>n(x)</>m(x)dx = 0 if m n; J[^.(^)]2 dx = Nn (I) and for sufficiently regular functions fx)y

00

f(X) = X Cn<l>nX) (2) n=l

where Cn = W Sf(x^(x) dx (3)

The integrals are all taken over the fundamental interval for the problem, and the expansion holds in this interval. Except for special conditions sometimes required at the ends of the interval, or at infinity when the interval is infinite or semi-infinite, the regularity conditions are usually similar to those for Fourier series.

It is possible to create a set of orthogonal functions cf)n(x) from any linearly independent infinite set fn(x) by setting

" 1 <£i(*0 = / i (*)> Hx) = hix) ~

1

N lHx)Ux)dx Hx) (4)

<l>n(x)=fn(x) - X $<t>iX)fnX)dx UX) (5)

Examples are Pn(x) on —1,1 of § 8.8 ; Pnm(x) for fixed mon —1,1 of § 8.16 ;

xll2Jp(avnx) where JâV7) = 0 for fixed positive^) on 0,1 of § 9.13 ; e~xl2Ln(x) on 0, oo of § 11.4 ; e- a j / 2»*+1Ln + f c

2*+1(jc) for fixed k on 0, «> of § 11.7; e-'^hEnx) on — o o , oo of §12.4; Vx^l — xy-«Jn(p9q\x) on 0,1 of §10 .7 ; 1/(^1 ~x*)Tnx) on - 1 , 1 of § 10.7.

14.20. Weight functions. The functions gn(x) are orthogonal with a weighting factor w(x) when

Mx)gnx)gm(x)dx = 0 if m^Ln (1)

For such functions the expansion is

m-=%cngn(x) ' (2) 71=1

where Cn = ~--Jw(x)F(x)gn(<x)dx, Mn = lw(x)[gnx)fdx (3)


Illustrations related to some of the examples mentioned at the end of § 14.19 are Jp(apnx) with w(x) = x; Ln(x) with w(x) = e~x; Hn(x) with w(x) = e~x*; Jn(p,q ;x) with w(x) = xq-\\ — x)*~q ; Tn(x) with w(x) = l/V 1 — x2.

15. Asymptotic Expansions

15.1. Asymptotic expansion. Let the expression

a) t/v I/V «/V

be related to a function F(x) in such a way that for any fixed n

a, a, lim * n

xa

= 0 (2)

The limit holds in some sector of the complex plane. The expression in general is a divergent series, since it can converge only if F(z) is analytic at infinity. In any case it is called an asymptotic expansion for the function. The terms asymptotic or semiconvergent series are also used.

The error committed when we employ a finite number of terms in place of f(x) is frequently of the same order of magnitude as the numerical value of the next following term. Thus asymptotic expressions can be used in computation like convergent alternating series as long as the terms remain small. For examples see §§ 8.9, 8.14, 9.7, 13.5, and 13.9.

If the function F(x) is an infinite series of terms 2 uk(x), an asymptotic series may sometimes be found by taking uk(x) as the f(k) in the Euler-Maclaurin sum formula of § 14.4.

15.2. Borel's expansion. Let CO

be any function for which the integral

I(x) = ê-txt^(t)dt (1)

converges. Then the expansion

is usually an asymptotic series for I(x). One extension is

m j y - - mt)dt=| An r ( £ + i) J L

(2)

(3)


15.3. Steepest descent. If the path C is suitably chosen in the complex plane (see § 20.3) the integral

F(x) = jce-x°wh(t)dt (1)

may be transformed into integrals whose expansion can be found by § 15.2. The path C is one along which g(t) remains real and changes most rapidly. It will pass through a saddle point of the surface R(u,v) where t = u + iv and g(t) = R(uyv) + il(u,v). At this saddle point t0i g(t0) = 0 and gr(t0) = 0, and expanding about it

* = g(t) = (t- taf[g0 + gl(t - tQ) + g2(t - t0f +...] (2)

The path C consists of two parts Lx and L2. On Ll9 we find t-^s) by integrating dtx between 0 and s to obtain tx — 1 0 . And on L 2 , we get t2(s) by integrating dt2 between 0 and s to obtain t2 — £0. Here

7. 1 CO 7 , -j CO

where in terms of the coefficients gki

Then in terms of s,

F(x) = f e-^hfa) ^r-ds+ f e-* sA(*2) ^ ^

Since s is real, if s runs from 0 to infinity on the path C = Lx + L 2 , the sum of these integrals in general has an asymptotic expansion which follows from Borel's expansion (2) of § 15.2.

16. Least Squares

16.1. Principle o f least squares. Let n > r, so that the n observation equations

a& *r) = sk, k= 1, 2, ... n (1) form an overdetermined system for the determination of the r unknown constants aq. Then when the n observed quantities sk have comparable accuracy, the kth. residual is taken as

fl2» •••> ar) — *k (2)

§ 1 6 . 2 BASIC MATHEMATICAL FORMULAS 81

The principle of least square asserts that the best approximation to the aq

is the set for which the sum of the squares of the residuals is a minimum. Necessary conditions for

S = (3)

to be a minimum are

dax ' da2 ' ' dar ^ ^

These constitute the normal equations. In general they determine a unique solution for aq which gives the desired best approximation.

16.2. Weights. When the relative accuracy of the sk in § 16.1 is known to be different, we assign weights wk such that the quantities wksk have comparable accuracy. Then the residual vk is taken as

vk = wkfk(alya2,...,ar)—wksk (5)

and otherwise the principle is applied as in § 16.1,

16.3. Direct observations. When the sk are direct observations of a quantity aly the residual vk is ax — sk, and the principle gives the average

a1 = ^-(s1 + s2 + ... + sn) (1)

as the best approximation for av

16.4. Linear equations. When linear in the aQ, the n observation equations may be written

r

^£fAkQaQ = ski k=l92,...yn (1)

In this case the normal equations are r n n

2) 2/ AkvAkqaQ — ^ Akvsky p=l,2, ...yr (2) 9=1 Jc=l k=l

16.5. Curve fitting. Suppose the n points (xk,yk) follow approximately a straight line. T o find the best line when we regard yk as the observed value corresponding to we write

a1 + a2xk=yk (1)


as the observation equation. And we determined a± and a2 by solving the two simultaneous normal equations

( n \ n

X x A a * = X^* fc=l I fc=l

X + X r 2 = X / \k=l I k=l

(2)

Similarly, to find the best parabola we use

«1 + a2Xk + #3Xk2 = Jfc (3)

as the observation equations, and proceed as in § 16.4, or we may use a polynomial of the wth degree, with the observation equations

«i + a&k + a&k + • • • + am+ixkm = Jk (4)

16.6. Nonlinear equations. When the functions fk are not linear in the aq, we get a first approximation dg by graphical means when r = 2, or by solving some set of r of the observation equations. Then we expand in Taylor's series about dQy and neglect second-order terms. The result is

fk(alya2,...yar) + ^•£-&ak = sk (1)

We treat these as observation equations in the unknown &ak. With

dfkld&q for ^/ca» A a a for a v $k —fkid^d^'-A) for

they have the form (1) discussed in § 16.4.

17. Statistics

17.1. Average. The average of a finite set of observed values sv s2, ...,sn is

fz1(s) = s = ^(s1 + s2 + ... + sn) (1)

17.2. Median. When a sequence is arranged in order of magnitude the number in the central position, or the average of the two nearest to a central position, is called the median. Essentially, there are as many numbers in the sequence larger than the median as there are smaller than the median.


17.3. Derived averages. The kth moment is the average of the kth powers so that

= ^ = - ^ ( * l * + * 2 * + + (1)

17.4. Deviations. T o the set of observed values sv s2i sn corresponds an average s, § 17.1, and a set of deviations

= — s, d2 = s2 — s, dn = sn — s (1)

The average of a set of deviations is zero, p^d) = <l — 0.

For the mean-square deviation § 17.3,

,*.(<*)=/*>(*)-fruW]* ™ ^ = ^ - ( s f (2)

The square root of n>2d), or root-mean-square value of the deviation is called the standard deviation cr.

o = VtW)= \ / ^ S ^ 2 (3)

17.5. Normal law. The deviations are samples of a continuous variable x, whose distribution function may frequently be taken as the normal law of error

y = -ê-^ ( 1 )

That is Pab the probability that an error is between a and b, is

In terms of the error function of § 13.9.

Pa,b = i (Erf hb- Erf ha) and P__ M = ErfA* (3)

For a given normal distribution, h is called the measure of precision.

17.6. Standard deviation. For the normal law of § 17.5 the second moment is

Thus the standard deviation is


^2k(x) = 1 • 3 • 5 • ... (2k - 1) * = 1 • 3 • 5 • ... (2k - i y * * (3)

17.7. Mean absolute error. For the normal law of § 17.5 the mean absolute error is

^ ( L - L ) = 2 j > ^ e - ^ = ^ = (1)

Thus the mean absolute error

1 • 1 0.5643 n . hy 77 /xv 7

17.8. Probable error. For the normal law of § 17.5 the particular error which is just as likely to be exceeded as not is called the probable error, e. Thus

\ = P_Ee = Erih€ and he = 0.4769 (1)

And the probable error

0.4769 0.6745a = 0.8453/x (2)

17.9. Measure of dispersion. To find a measure of precision h which makes the normal law fit a set of deviations, we identify the standard deviation of § 17.6 with Vn$n — 1) times that of § 17.4. Thus

Or, we may identify the mean absolute error of § 17.7 with Vnl(n-I) times that formed from the dr, ^(1 dr$. Thus

1 1 v , , , , 0.8453 v | . . n . hy 77 r yn(n — \) 1 r i yn(n — \) 1 1 v /

These are easier to calculate, but slightly less accurate, than the values found from the standard deviation. In some cases both values are found, and a rough agreement is considered a check on the normal character of the distribution. In either case the probable error of the mean is taken as times that for a single observation, em = e/y^n.

17.10. Poisson's distribution. A variable taking on integral values

All the odd moments, fJL2k+i(x) = 0, and for the higher even moments 1

2Wk


o n l y > k = 1, 2, 3, ... has Poisson's distribution law if the probability that the variable equals k is

aKe~ M W

For this distribution the average value of the variable

Pi(k) = a (2) and the standard deviation

CJ = Vfjb2(k) = X/a, a = a 2 (3)

These relations may be used to find the parameter a from the average or standard deviation of a sample set of observations which follow this law.

17.11. Correlation coefficient. Let x and y be two variables for each of which the deviations from the average have a normal distribution. Then for a set of corresponding pairs xlyyx \x2yy2 ;xnyyn the correlation coefficient of the sample is

S f o - f f ) (yt-y)

where

1 y = t*i(y) = — = yi-y

(i)

(2)

(3)

For r = 0 there is no correlation. For r = 1 there is strict proportionality y = Cx with a positive C, and for r = — 1 with a negative C. For | r \ near 1, the ellipses of equal probability for the distribution of points xhyi are long and thin, approximating a straight line. The constant c determined from

2S^-D, tan 2cf) • c = tan cf) (4)

leads to the straight line y = cx such that the sum of the squares of the distances to all the observed points xiyyi is a minimum.

18. Matrices

18.1. Matrix. The elements of a matrix are a system of mn numbers aik(i = l ,2, . . . ,m; k = l,2,...,w). These form a matrix when written as a rectangular array of m rows and n columns

(i)


We use the abbreviations || aik || or a single letter A. The double bars are sometimes replaced by parentheses. (See §§ 7.4, 7.5.)

18.2. Addition. The relation

A = B means aik = bik (1)

A + B = C means aik + bik = cik (2)

sA = A means saik = bik9 where s is a scalar (3)

In each of these relations all the matrices involved must have the same dimensions. If these are m by n the matrix equation is equivalent to mn scalar equations.

18.3. Multiplication. Two matrices A and B, taken in this order, are said to be conformable if the number of columns of A equals the number of rows of B. Let A be m by N and B be N by n. Then the product C = AB is an m by n matrix with

N

3=1

If n ^ m, then B and 4 will not be conformable, and there is no product BA. If A is m by w and B is w by w, then AB is an /n by m matrix and BA is

an n by w matrix, of different dimensions if m ^£ When 4 and B are each w by n, or square matrices of order ny the product

AB and iL4 will each be n by w, but in general will not be the same. Matrix multiplication is not commutative.

Matrix multiplication is associative,

A(BC) = (AB)C = ABC (2)

18.4. Linear transformations. Let X\ — || xk \ \ be a 1 by n column matrix, Y\ = \ \ yi \ \ be a 1 by iV column matrix, and Z\ = 11 zt \ \ be a 1 by m column matrix. Then if A is m by N, and B is Ar by n, in matrix form the transformation

y,= %h**k ^ Y\=BX\ (1) fc=l

and the transformation

* < = i > ^ ^ Z\=AY\ (2)


AX\=B\ (2)

If the transformation which takes the xk into the zi directly is

*i = %cikXk is Z\ = CX\, then C = AB and Z\ = ABX\ (3) k=l

18.5. Transposed matrix. If A = || ^ || is w by N, the transposed matrix A' = || 0^ || is Af by m. For a product C = AB, the transposed matrix is C = B'A'.

For X\, Y\y Z\ the column matrices of § 18.4 with a single column, the transposed matrices will be row matrices with a single row. We denote them by X, Y, Z. Then the transformations of § 18.4 may be written in terms of row matrices as

Y = XB\ Z = YA', Z = XC\ or Z = XB'A', since C = B'A'

18.6. Inverse matrix. The unit matrix is a square matrix with ones on the main diagonal and the remaining elements zero. Its elements are &ik = 0 ^ i ^ k, and = 1 if i = k.

A square matrix is singular if its determinant is zero. Each nonsingular square matrix has a reciprocal matrix A-1 such that

^ - i = ||8tt|| and A-iA = \\8ik\\ (I)

Let | A | denote the determinant | aik |, and Aik the cofactor of aik or product of (— l ) i + k by the determinant obtained from | aik \ by striking out the z'th row and &th column. (See §1.8.) Then explicitly A_1 has as elements

a~\u = | ^ (2)

18.7. Symmetry. For square matrices symmetry and skew-symmetry are defined as for tensors in § 7.10. Orthogonal and unitary, or Hermitian orthogonal matrices are defined in § 7.11.

18.8. Linear equations. The set of n linear equations in n unknowns xk,

n

X ai*xk = hv i=l,2,...,n (1)

determine a square matrix A = • 11 aik 11 and two column matrices B\ — || B2: || and X\ = \ \ xi ||. In matrix form we may write


If A is nonsingular, find its solution by premultiplying by the inverse matrix A-1 of §18.6.

i n X\=A-*B\ or x i = " Âkibk (3)

I ^ I k=l

18.9. Rank. For a rectangular matrix we get a subdeterminant of the qth order by selecting those elements in some set of q rows and some set of q columns. If at least one subdeterminant of order r is not zero, but all the subdeterminants of order r + 1 and hence those of higher order vanish, the matrix is of rank r.

With any set of m linear equations in n unknowns xk,

n

X = *<» i = !>2,...,m (1) k=l

or AX\=B\ with ^ = ||ait||, X\ = \\xk\\, B\ = \\bi\\ (2)

we associate two matrices. The matrix of the system is the m by n matrix A, and we obtain from A the augmented matrix, which is an m by (n + 1) matrix, by adding an m + l)st column whose elements are those of B\. Then the condition that the system of equations is consistent in the sense of having one or more sets of solutions for the unknown xk is that the matrix and augmented matrix have the same rank.

18.10. Diagonalization of matrices. Let A = \ \ aH 11 be a square n by n matrix, and || S - || be the unit matrix of § 18.6. Then || AS - — aj || is the characteristic matrix of A. The roots of the characteristic equation, | Aha — ai5 | = 0 are the eigenvalues of A. Suppose that they are all distinct, Ax, A2, Xn. Find solutions of

aikXkj ~ Xjxij'

The ratios of the xijy for each j , are determined, § 18.9. They may be scaled to make the vector xljy x2j, xnj have unit length. For distinct Xjy

any two of these vectors are orthogonal. The orthogonal (§7.11) matrix X— || xj || is such that X_1AX = || A^S^ ||, a diagonal matrix.

When the eigenvalues are not all distinct, a reduction to diagonal form may not be possible, and we are led into the theory of elementary divisors.

Whether all eigenvalues are distinct or not, whenever A is symmetric, § 7.10, unitary, § 7.11, or Hermitian, H= H, a unitary matrix U can be found


such that U~YAU = || A4-8# ||, a diagonal matrix. Here U = || w# [|, where if is a root of | AS - — | of multiplicity m, so that A - = A m = ... = A^+w-!, the columns of uj from j to j + m — 1, uiSJ are found from the m linearly independent vector solutions of

n

aijxkt = ^'î*

We take ^ =

where the bar means complex conjugate

Vtt = xit — X ~W~ (X fc ptVifc 7c=i ' k \ p=i J

Here n

Vk = X Then « < i i + M = ( l / V ^ ) » « .

The eigenvalues of a real symmetric or of a Hermitian matrix, H = H, are all real. The eigenvalues of a real orthogonal or of a unitary matrix are all of absolute value 1, dz 1 o r ei^>

19. Group Theory

19.1. Group. Let a rule of combination be given which determines a third mathematical object C from two given objects A and J5, taken in that order. We call the rule " multiplication " ; call C the " product," and write C = AB. Then a system composed of a set of elements A, B, and this one rule of combination is called a group if the following conditions are satisfied.

I. If A and B are any elements of the set, whether distinct or not, the product C = AB is also an element of the set.

II. The associative law holds ; that is if A, By C are any elements of the set, (AB)C = A(BC) and may be written ABC.

III. The set contains an identity or unit element I which is such that every element is unchanged when combined with it,

IA = AI=A (1)

90 BASIC MATHEMATICAL FORMULAS. § 1 9 . 2

IV. If A is any element, the set contains an inverse element A~\ such that

A-1 A = AA-1 = I (2)

19.2. Quotients. For any two elements A and B there is a left-hand quotient of B by A, such that AX = B. This X = A~XB. There is also a right-hand quotient of B by A, such that YA == B. This F = BA~X. Let be a fixed element of the set, and V be a variable element taking on all possible values successively. Then V~x, VF> FV, V~XF, and FV~X each runs through all possible values.

19.3. Order. If there is an infinite number of elements in the set, the group is an infinite or group of infinite order. If there is a finite number, g, of elements in the set, the group is a finite group of order g.

19.4. Abelian group. If the rule of combination is commutative so that all cases AB = BA, the group is an Abelian group. In particular a finite group whose elements are all powers of a single element A, such as A, A2, A3, A9 = I is necessarily Abelian and is called a cyclic group.

19.5. Isomorphy. We use a single capital letter to indicate either a group or its set of elements. Two groups G and G' are isomorphic if it is possible to establish a one-to-one correspondence between their elements G and G' of such a sort that if A, B are elements of G and A\ B\ are the corresponding elements of G', then AB corresponds to A'B'.

If it is possible to establish an ra-to-one correspondence between the elements G and G' of this sort, the group G is multiply isomorphic to the group G'.

19.6. Subgroup. Let G be a finite group of order g. Let the elements H be a subset of the elements G such that if AH and BH are any two elements in H, their product AHBH is also in H. Then the system composed of the elements H and the rule of combination for G is a group in the sense of § 19.1. We call this group H a subgroup of the group G. The order of the subgroup H, hy is a divisor of g. And the integral quotient j = gjh is called the index of H with respect to G.

19.7. Normal divisor. The elements A~XHA, where A is any fixed element of G and H takes on all the elements of some subgroup H, themselves form a subgroup of G, which is said to be conjugate to the subgroup H. If,


as A takes on all possible values as an element of G, the resulting conjugate subgroups of H are all identical, then the subgroup H is called a normal divisor of the group G.

19.8. Representation. Let M be a set of /z-rowed square matrices, each of which is nonsingular, which form a group when the rule of combination is matrix multiplication as defined in § 18.3. If the group M is simply or multiply isomorphic to a group G, the group M is called a representation of G in terms of matrices or linear transformations, § 18.4. The number of rows in the matrix, or of variables in the linear transformations, h, is called the degree or dimension of the representation.

Two representations M' and M are equivalent if for some fixed matrix A, M' = A_1MA. If for some p + q = h, each matrix M consists of two square matrices of p and q rows, respectively, on the main diagonal, surrounded by a p by q and a q by p rectangle of zeros, the representation M is reducible. Otherwise it is irreducible. For each finite group G of order g there are a finite number c of irreducible representations not equivalent to one another. If these have dimensions hlf h2, hc, each of the hn is a divisor of g, and

V + V + - + K* = g

19.9. Three-dimensional rotation group. The elements of this infinite group are real three by three orthogonal matrices, § 7.11, B — || bj ||. As in § 18.4, Y = BX, where x±, x2, x3 = x, y, z of a first coordinate system and y±, y2, y3 = x', y', z' of a second rotated coordinate system. The particular matrices

| cos a sin a 0 Rz(oc) = 1 — sin a cos oc 0

J O 0 1

correspond to rotations about OZ through oc and about OF through j8. For a suitable choice of the Euler angles a, j8, and y any matrix of the rotation group B = R(ot,p,y) = Rz(y) Ry(p) Rz(oc).

Consider next the unimodular matrices of order two, or unitary matrices with determinant unity,

E7=|| a + bt £+f.\\ (2) II — c + ai a — bi II

where a2 + b2 + c2 + d2 = 1. As in § 7.11, Uvv = U-\q. These make up the special unitary group.

cos jS 0 — sin j8 ^ 0 8 ) = || 0 1 0

sin /3 0 cos f3 a)


The Pauli spin matrices are defined by 0 1| 1 0 I , ^ 2 =

0 M l p 0

-i oil' 3_L!o - l

(3)

= H(x,y,z) = xP1 + yP2 + zP3 (4)

The Hermitian matrix with x2 + y2 + z2 = 1, || z x ^ iy 11 * — iy —z

has. H = H. And for any unimodular matrix C7, if

U-iHU = x'Px + y'P2 + * 'P 3 ,

the transformation from x,yyz to x\y\z' is a three-dimensional rotation. The rotations RZ(OL) and P^/?) correspond to Uôc) and £/2(/?) where

J3 „:„ J3 c<«/2 0 COS

0 e"^ ' 2 sin

(5)

Thus R(<x,p,y) correspond to £/(a,)3,y) = l ^ y ) £/2(/3) t/â), or

2 " 2 e i ( « + y ) / 2 C 0 S A _ e - t ( « - r ) / 2 s m

e t ( « - y ) / 2 S M A . e - i ( « + y ) / 2 C O S A

(6)

The irreducible representations, § 19.8, by matrices of order (2j + 1), of the special unitary U group are given by

* V(j+P)\ (J -P)\ (j + g)i (j - g)i (j — p — m)! (; + — w)! (m Ar p — q)\ ml

> (?)

X 9 a COS' )8 • sin p+2m— # _L__ pipy

Here >,# = —/ , — j + 1, ... ,j — l,jy and m = 0, 1,2, ... where we stop the summation by putting l/(—N)l = 0 ; andj = 0, \, 1, §, 2, ... .

The group characters, or traces, § 7.12, of these matrices are sin + \)OL

XU)((x) = 1 + 2 cos (x + ... + 2 cos joe =' sin a/2 (8)

f o r ; = 0, 1,2, 3, ..


F o r ; = 1 , 1 , 1 , . . . ,

x «> ( « ) = 2 cos -f- + 2 cos ^ + ... + 2 cos /a = ( 9 )

A, sin oCj Each matrix R(ocyp,y) corresponds to two U matrices, namely, £/(&,/?,/) and

U(OL + 277, j8, y). The group R(ocfi,y)U(oifi,y) is simply isomorphic with U(oc$,y) and has all the matrices c T ^ ^ given above as representations.

But for the group R(oifi,y) the matrices c T ^ ^ and characters xU)(oc) with y = 0, 1, 2, 3, ... give a complete representation.

20. Analytic Functions

20.1. Definitions. Consider the complex variable z = x + iy where i2 = — l. We associate z=*x-\-iy with the point (x,y). The single-valued function f(z) has a derivative //(<2') a t # if

/ ' ( * ) = Hm ^ * + A * > V Aa-.0 #

where A#—• 0 through any complex values. Let z be a variable and z0 a fixed point in R, any open (boundary excluded)

simply connected region. The function / ( s ) is analytic in R if and only if any of the following four conditions hold.

a. f(z) has a derivative/'(#) at each point of R.

b. f(z) may be integrated in R in the sense that the integral ff(z) dz = 0 about every closed path in R. Thus

F(z) = \Z fz)dz

is a single-valued analytic function of z independent of the path in R.

c. f(z) has a Taylor's series expansion, §§ 3.25, 20.4, in powers of (z — z0) about each point z0 in R.

d. f(z) = u(x,y) + iv(x,y), where both u(x,y) and v(x,y) have continuous partial derivatives that satisfy the Cauchy-Riemann differential equations,

du dv du dv m

dx dy' dy dx

The functions u and v are conjugate potential functions, and each satisfies Laplace's equation

— + — - 0 (2)


20.2. Properties. If f(z) is analytic at all points of some circle with center S, but not at S, then z0 = S is an isolated singular point. It is a pole of order n if n is the smallest positive integer for which (z— Zo)nf(z) is bounded. If there is no such n, then z0 is an essential singularity.

The function f(x + iy) = u(x,y) + iv(x,y) effects a mapping from the x,y to the u,v plane. At any point where f'(z) ^£ 0, the mapping is conformal, preserving angles of image curves and shapes and ratios of distances for infinitesimal figures.

20.3. Integrals. The integral

j 1 ' f(z) dz = j b (u + iv) (dx + i dy) \

(i) = j (u dx —v dy) + i' j (v dx + u dy) J

Each of the real differentials (u dx — v dy) and (v dx + u dy) is exact. The integral

is not changed when the path from a to b is continuously varied without crossing any singular point. And

j>f(z)dz = jaJ(z)dz = 0

over any closed path not enclosing any singular point. For the same type of path enclosing t, traversed in the positive sense, Cauchy's integral formula asserts that

™ = h$irk*' ( 2 )

20.4. Laurent expansion. Let f(z) be analytic for 0 < | z — z0 | < r. Then for z in this range,

oo

/(*)= X • s„)" ( 1 )

••• + , U~l v + + a 0 + a 1 z - z 0 ) + a i ( z - z 0 f + ..

(Z — Z0) z — Z0 /

a m = 2^H$c(z- zo)-™-1/^ (2)

where C is a circle | z — z0 \ = a with a < r traversed in the positive direc-


tion. When f(z) is analytic at z0, all the am with negative subscripts are zero, and the expansion reduces to Taylor's series, § 3.25, with an = (l/n!)/<«>(«).

When f(z) has a pole of order n at z0, § 20.2, the am are zero for m < — n. In this case the residue a-x may be found from

In particular for a pole of the first order, n = 1 , the residue a-± may be found from

a _ 1 = lim [ ( * - * „ ) / ( * ) ] ( 4 ) Z-*ZQ

For an isolated essential singularity, § 20.2, there are an infinite number of negative powers in the expansion, and we still call the residue.

20.5. Laurent expansion about infinity. Similarly let f(z) be analytic for | z | > r. Then for z in this range,

/(*) = X *™*m> "m = ^j*rm-1f(*)d* (1)

where C is a circle | z | = # with a > r traversed in the positive direction. In this case for z = <», f(z) is analytic if there are no positive powers, f(z) has a pole of order n if all the <zw are zero for m > w, and/(^) has an isolated essential singularity if there are an infinite number of positive powers.

20.6. Residues. About a circle C, | z — z0 | = a as in 2 0 . 4 ,

k ( * - * o ) w < k = = 0 ( 1 )

when m — 0 , 1, ± 2 , ± 3, except for m = 1, when

(f — ^ — = 277/ ( 2 ) J c — z0

v ;

Term wise integration of the Laurent series of § 20 . 5 over C gives §cf(z) dz= 2ma_x. And about any closed contour passing through regular points only, but enclosing singular points zky §f(z) dz = 27ri[HR(zk)], where R(zk) is the residue a,-x or coefficient of l/(z — zk) in the Laurent expansion appropriate to zk. For poles the residues may be found as indicated in § 2 0 . 4 .


For a number of examples of the use of this residue theorem to compute real integrals see Franklin, Methods of Advanced Calculus, Chap 5.

21. Integral Equations

21.1. Fredholm integral equations. Fredholm's integral equation (F) is

U(x) = f(x) + A jba K(x,t)U(t)dt (1)

The kernel K(x,t) and/(#) are given functions, and U(x) is to be found. We assume that

jba [K(x9t)]2 dxdt=W

is finite. With (F) we associate the homogeneous equation (Fh), which is

U(x) = X jbaK(xyt)U(t)dt

and the transposed equation (Ft), which is U(x) = f(x) + A K(t,x) U(t)dt.

The homogeneous transposed equation (Fht) is

U(x) = âK(t,x)Ut)dt.

The number p of linearly independent solutions u±(x), u2(x), uQ(x) of (Fh) is finite and equal to the number of linearly independent solutions ui(x)> u2(x)> •••> UQ(X) ° f (Fht)' The number p is the defect of the kernel K(x,t) for the value A. The general solution of (Fh) is

ciui(x) + c2u2(x) + ... + ceue(x).

If p = 0, then (F) and (Ft) each has a unique solution for any f(x). There is a solving kernel, or resolvent, T(x,t ;A) such that

U(x)=f(x) + \jbar(x,t;X)f(t)dt for(F),

and

U(x) =f(x) + XJba T(t,x \X)f(t)dt for (Ft).

Fredholm's form of the resolvent is

IX*,';A) = ^ (2)

§ 21.2 BASIC MATHEMATHICAL FORMULAS 97

where oo An

D(x,t;X) = K(x,t) + X (-l)nDn(x,t) —

and

= 2 ( - 1 ) B A A

The Z)'s are found in succession from

Dm = \h Dn^xrfdx, Dm(x,t) = DmK(x,t) —m\h Kx,s)Dm_Y(s,t)ds.

These series converge for all A. For sufficiently small A, we have the Neumann series

T(x,t;A) = K(x,t) + mix,t) + \*Ks(x,t) + ... + XnKn+1(x,t) + ... (3)

The iterated kernels Km(x,t) are found in succession from

K^xj) = K(x,t) and Km(x,t) = KfaijKm^sfids (4)

If p > 0, then (F) has a solution only it

jbaUi(t)f(t)dt = 0

for z = 1,2, The general solution of (F) is one solution plus the general solution of (Fh).

21.2. Symmetric kernel. In §21.1 let K(x,t) = K(t,x). Then (Fh) has nonzero solutions for certain determined discrete values Al 5 A2, Aw,

called the eigenvalues, for which the defect p > 0. The corresponding solutions u(x) of (Fh) are the eigenfunctions. For any A, p ^ A2 IF. For IF see §21.1.

Every symmetric kernel, not identically zero, has at least one and at most a denumerable infinity of eigenvalues, with no finite limit point. These eigenvalues are all real if K(x,t) is real and symmetric. At most A2 W eigenvalues have numerical values not exceeding A, and A n

2 ^ \jW. Also

oo i

where each eigenvalue of defect p is counted p times. The p functions for a A of defect p may be taken as of norm, the Nn of

§ 14.19, unity, and orthogonal to one another. Then if the functions for a A


where fn = jbJt)un(t)dt.

(6)

Any function which is sourcewise representable,

F(x) = âK(x,t)<t>(t)dt (7)

has an absolutely and uniformly convergent series development

FX) = %FNUNX) where FN = ^FFFYUJFYDT. (8) 71=1

2 1 3 . Voltcrra integral equations. Volterra's integral equation

(V) is U(x)=f(x)+XJXQK(x,t)U(t)dt (1)

This always has a solution

U(x)=f(x) + XJXoT(x,t;X)f(t)dt (2)

where the resolvent

T(x,t ;A) = K(x,t) + XK2(x,t) + \*K3(x,t) + ... + \nKn+1(x,t) + ... (3)

with p = 1 are taken as of norm 1, the totality of eigenvalues will form a real normal and orthogonal set as in § 14.19. Thus

j b [un(x)]2 dx = I, j b un(x)um(x)dx = 0, m ^ n.

The bilinear series

KX,T) = % U ^ F ^ ( 5 ) 71=1 XN

converges for a Mercer kernel with only a finite number of positive, or only a finite number of negative eigenvalues. Otherwise it converges in the mean,

lim j(K-Snfdx = 0

where Sn is the sum to n terms.

If (F) has a solution, it is given by the Schmidt series

fn


The iterated kernels Km(xyt) are found in succession from

Kxx,t) = K(xyt)y Km(xyt) = j^Kix^K^s^ds (4)

This Neumann series converges for all values of x.

If H(xyx) ^£ 0, we may reduce the solution of the equation of the first kind,

g(x) = A jXoH(x,t)Ut)dt (5)

to the solution of an equation of type (V). Differentiation with respect to x leads to

g'(x) = ^jl^ U(t)dt + XH(xyx)U(x) (6)

This is of type (V) with *

J y ' AH(xyx) v ' AH(xyx) dx

21.4. The Abel integral equation. AbePs equation is

2(x) = \*a(^yfdt> w i t h 0 < ? < i and g(a) = 0 (1)

For this kernel, singular at t = xy the unique continuous solution is

(t — x)1-" (2)

21.5. Green's function. Let L(y) = E(x) be the general linear differential equation of the nth order (4) of § 5.13. The solution satisfying given boundary conditions may often be written as

y(x) = jbaG(xyt)E(t)dt (1)

where G(xyt) is the Green's function for the given equation and boundary conditions. This function satisfies L(G) = 0 except at x = t. At x = ty

G(xyt) as a function of x is continuous, together with its first (n — 2) derivatives, but

dn~xG \x=t+ 1 dx"1-1 \x=t- ~~ An

where, as in (4) of §5.13, An is the coefficient of dny\dxn in L(y). The Green's function G(xyt) must also satisfy the same given boundary conditions as were imposed on y(x).


The solution of the differential equation L(y) + Xw(x)y = F(x) satisfying the given boundary conditions satisfies the Fredholm equation (F) of § 21.1

y(x) = X G(x,t)w(t)y(t)dt + G(*,*)F(*)<# (2)

If [vL(u) — uL(v)]dx is an exact differential, L(y) is self-adjoint, and the Green's function, if it exists, is necessarily symmetric, so that G(x,t) = G(t,x). In this case let

U(x) =y(x)Vfo(x)9 f(x) = \/w(x) G(x9t)F(t)dt

Then (3)

U(x) = A jba G(x,t)V^Mt)U(t)dt + /(*)

which is an integral equation with symmetric kernel like (F) of §21.2 with K(x9t) = G(x,t)Vw(x)w(t).

All the conclusions of § 21.2 apply, and in particular any n times differen-tiable function satisfying the boundary conditions can be developed in a series in terms of the un(x). Hence there are an infinite number of eigenvalues and the eigenfunctions form a complete set.

21.6. The Sturm - Liouville differential equation. If L(u) = (pu'Y — qu, then

[vL(u) — uL(v)]dx = d[p(vu' — uv')] (1)

Thus the Sturm-Liouville equation

L(u) + Xwu = 0, or (pu'Y —qu + Xwu = 0 (2)

is self-adjoint, and the theory of §§ 21.5 and 21.2 applies to it. We describe a number of important particular examples. We write p(x)

and q(x) in place of p and q when p and q are used as parameters with other meanings.

a. Let the boundary conditions be u(—1) and u(\) finite. With p = 1 —x 2 , q == 0, w = 1, we have the Legendre equation of § 8.1 with eigenvalues n(n + 1) and eigenfunctions the polynomials Pn(x) of §8.2. With

tn2

p = l - x 2 , q= 1 w = 1 (3)

we have the associated Legendre equation of § 8.10 with eigenvalues n[n -f 1), n ^ my and eigenfunctions the polynomials Pn

m(x) of § 8.11. With

p(x) = xq(l — x)p-*, q(x) = 0, w(x) = JC«-x(1 — x)*~q (4)


we have the Jacobi equation of § 10.7 with eigenvalues n(p + n) and eigen-functions the polynomials Jn(p,q ;x). With

p = (\-x2)ll2> q = 0, w = (I - x2)"112 (5)

we have the Tschebycheff equation of § 10.7 with eigenvalues n2 and eigen-functions the polynomials Tn(x).

b. Let the boundary conditions be u(0) finite, and appropriate behavior at plus infinity. With

p = xe~x, q = 0, w = e~x (6)

we have the Laguerre equation of § 11.1 with eigenvalues n and eigenfunc-tions the polynomials Ln(x). With

p = x, q = -|- - y , w = 1 (7)

we have the Laguerre equation of § 11.4 with eigenvalues n and eigenfunc-tions e~xf2Ln(x). With

p = x8~xe~x, q = 0, w = xse~x (8)

we have the associated Laguerre equation of § 11.5 with eigenvalues r — s and eigenfunctions the polynomials Lr

s.

c. Let the boundary conditions be appropriate behavior at plus infinity and at minus infinity. With

p = e~x\ q = 0, w = e~x2 (9)

we have the Hermite equation of § 12.1 with eigenvalues In and eigenfunctions the polynomials Hn(x). With

p = 1, q = x 2 - l , .w = 1 (10)

we have the Hermite equation of § 12.4 with eigenvalues 2n and eigenfunctions e~x2/2HJx).

d. Let the boundary conditions be w(0) finite, u(l) = 0. With n2

p = x, q = —, w = x, A = a2 (11) x

we have an equivalent of the modified Bessel equation

x2u" + xu' + (— n2 + a2x2)u = 0,

which is satisfied by Zn(ax) by §§9.5 and 9.6. The eigenvalues are an2y

where Jn(anr) = 0, and the eigenfunctions are the Jn(anrx), § 9.2. Here n is

102 BASIC MATHEMATICAL FORMULAS §21 . 7

fixed, and the eigenvalues and eigenfunctions correspond to r = 1, 2, 3, ... . For

p = xc(\ — »)°+b+1-c, q = 0y w = x ^ l — x)a+b~c, A = — ab (12)

the Sturm-Liouville equation reduces to the hypergeometric equation of §10.1.

21.7* Examples of Green's function. In each of the following examples, the Green's function G(x,t) = G^Xyt) for x ^ t and G(xyt) = G-^tyX) for x ^ t. Here Glm means a modified Green's function.

a. Let the boundary conditions be w(0) = 0, u(l) = 0.

For L(u) = w", Gâ;,*) = (1 — 0* (1)

For L(u) = u +k*Uy G1(xyt) = k~sm\ ( )

, 2 N sinh sinh k(l — t) For Lu) = u -k*Uy G1(xyt) = fe ^ ^ v ^ (3)

b. Let the boundary conditions be u(—1) = w(l), M'(—1) = u\\).

For L(«) = « ", Gln(*,t) = \ (* - 0 2 + y (* - t) + -g- (4)

For L(«) = a" + k*u, G^t) = - — (5)

For 2 * 0 = G L ( , F 0 = 5 ! ! ^ ^ ( 6 )

c. Let the boundary conditions be u(0) = 0, u\\) = 0.

For Lu) = u"y G1(x,t) = x (7)

T- 7-/ \ // , 7 9 A>/ .\ S m kX COS &(1 — / o x

For L(u) = u + klUy Gx(Xyt) = , ^ (8) K COS fZ

t? r/ \ " L9 ^ / .\ S M N kx COsh &(1 — 2) For L(u) = u — &2«, GAxyt) = — —-^ - (9)

v & cosh 6 J

d. Let the boundary conditions be u—1) = 0, w(l) = 0.

For L(M) = w \ G±(Xyt) = y (1 + x — t — xt) (10)


e. Let the boundary conditions be

i < 0 ) = - u ( l ) , u'(0) = -u'(l). (11)

For L(u) = u", G1(*,0 = - ^ + ^ - ^ (12)

f. Let the boundary conditions be w(+°°) and u(— OO) finite.

For L(u) = u"-u, Gx(*,0 = ( 1 3 )

g. Let the boundary conditions be u(0) = 0, w'(0) = 0, «(1) = 0, u\\) = 0. For L(w) = G^*,*) = ^ x\\ - *)2(2atf + x-3t) (14)

h. Let the boundary conditions be u(—1) and w(l) finite. For L(w) = [(l - * > ' ] ' ,

G ^ , 0 = - - i In (1 - *) (1 - t) + In 2 1

With this L(w), Legendre's equation of § 8.1 is

L(y) + *(* + % = 0

For L(u) = [(1 - x2)u'] 1 - * 2

(1 + * ) ( ! - 0'

, m ^ 0 ,

m/2 2m [(I - * ) 0 + 0J

With this L(w), the associated Legendre equation of § 8.10 is

L(y) + n(n + l)y = 0

i. Let the boundary, conditions be «(0) finite, u(l) = 0.

For L(u) — (#« ' ) ' , G^*,*) = — In x

With this Bessel's equation (3) of § 9.1 for n = 0 is

+ xy = 0 n2 1

For L(u) = (*u')' — — u, n ^ 0, G-^*:,*) = —

With this L(u), Bessel's equation (3) of § 9.1 is

*)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

L(y) + xy = 0 (23)


Bibliography

L Algebra

FINE , H. B., College Algebra, Ginn & Company, Boston, 1905. RICHARDSON, M., College Algebra, Prentice-Hall, Inc., New York, 1947.

2* Trigonometry and space geometry

BRENKE, W . C , Plane and Spherical Trigonometry, The Dryden Press, Inc., New York, 1943.

FINE , H. B. and THOMPSON , H. D . , Coordinate Geometry, The Macmillan Company, New York, 1918.

3. and 4. Calculus

DE HAAN , B., Nouvelles tables d'integrates definies, P. Engels, Leyden, 1867. DE HAAN , B., Tables d>integrates definies, P. Engels, Leyden, 1858-1864. D W I G H T , H. B., Tables of Integrals and Other Mathematical Data, The Macmillan

Company, New York, 1947. FINE , H. B., Calculus, The Macmillan Company, New York, 1937. FRANKLIN , P., Differential and Integral Calculus, McGraw-Hill Book Company, Inc.,

New York, 1953. FRANKLIN , P., Methods of Advanced Calculus, McGraw-Hill Book Company, Inc.,

New York, 1944. PIERCE, B. O., A Short Table of Integrals, Ginn & Company, Boston, 1929. WIDDER , D. V., Advanced Calculus, Prentice-Hall, Inc., New York, 1947.

5. Differential equations

FORSYTH, A . R . , Treatise on Differential Equations, Macmillan and Co., Ltd., London, 1885. (Dover reprint)

INCE, E. L., Ordinary Differential Equations, Longmans, Green & Co., London, 1927. (Dover reprint)

MORRIS , M. and BROWN , O. E., Differential Equations, 3d ed., Prentice-Hall, Inc., New York, 1952.

6. Vector analysis

PHILLIPS , H. B., Vector Analysis, John Wiley & Sons, Inc., New York, 1933. 7* Tensors

BRAND , L., Vector and Tensor Analysis, John Wiley & Sons, Inc., New York, 1947. EDDINGTON, A . S., The Mathematical Theory of Relativity, Cambridge University

Press, London, 1930. EISENHART, L. P., Riemannian Geometry, rev. ed., Princeton University Press, Prin

ceton, 1949. K R O N , G., Short Course in Tensor Analysis for Electrical Engineers, John Wiley & Sons,

Inc., New York, 1942. (Dover reprint) M C C O N N E L L , A . J., Applications of the Absolute Differential Calculus, Blackie & Son,

Ltd., London, 1931. (Dover reprint) W I L L S , A . P., Vector and Tensor Analysis, Prentice-Hall, Inc., New York, 1938.

(Dover reprint)


8. to 15. Special functions

CARSLAW , H. S., Theory of Fourier Series and Integrals, Macmillan and Co., Ltd., London, 1 9 2 1 . (Dover reprint)

COURANT , R. and HILBERT , D., Methoden der mathematischen Physik, Vol. 1, Julius Springer, Berlin, 1 9 3 1 . (An English translation, Methods of Mathematical Physics, is available from Interscience Publishers, Inc., New York, 1 9 4 3 .

FRANKLIN , P., Fourier Methods, McGraw-Hill Book Company, Inc., New York, 1 9 4 9 . (Dover reprint)

GRAY , A., MATHEWS , G. B. and MACROBERT , T. M., A Treatise on Bessel Functions, Macmillan and Co., Ltd., London, 1 9 3 1 .

HOBSON , E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, London, 1 9 3 1 .

K N O P P , K . , Theory and Application of Infinite Series, trans. Young, R. C , Blackie & Son, Ltd., London, 1 9 2 8 .

M C L A C H L A N , N. W., Bessel Functions for Engineers, 2d ed., Oxford University Press^ Oxford, 1 9 5 5 .

MARGENAU , H. and MURPHY , G. M., The Mathematics' of Physics and Chemistry, D. Van Nostrand Company, Inc., New York, 1 9 4 3 .

WATSON , G. N., Theory of Bessel Functions, Cambridge University Press, London, 1 9 4 4 .

WHITTAKER , E. T. and WATSON , G. N., A Course of Modern Analysis, Cambridge University Press, London, 1 9 2 7 .

18. Matrices

BOCHER, M., Introduction to Higher Algebra, The Macmillan Company, New York, 1 9 0 7 .

DICKSON , L. E., Modern Algebraic Theories, Benjamin H. Sanborn and Co., Chicago, 1 9 2 6 . (Dover reprint)

FRAZER, R. A., DUNCAN, W . J. and COLLAR , A. R., Elementary Matrices, Cambridge University Press, London, 1 9 3 8 .

TURNBULL , H. W . and AITKEN , A. C , The Theory of Canonical Matrices, Blackie & Son, Ltd., London, 1 9 3 2 .

19. Groups

SPEISER, A., Theorie der Gruppen von endlicher Ordnung, Julius Springer, Berlin, 1 9 2 7 . W E Y L , H., The Classical Groups, Princeton University Press, Princeton, 1 9 3 9 . W E Y L , H., Theory of Groups and Quantum Mechanics, Methuen & Co., Ltd., London,

1 9 3 1 . (Dover reprint) WIGNER , E., Gruppentheorie und ihre Anwendung auf Quantenmechanik der Atom-

spektren, F . Vieweg und Sohn, Braunschweig, 1 9 3 1 .

20. Analytic functions HURWITZ , A. and COURANT , R., Vorlesungen iiber allgemeine Funktionentheorie und

elliptische Funktionen, Julius Springer, Berlin, 1 9 2 5 . PIERPONT, J., Functions of a Complex Variable, Ginn & Company, Boston, 1 9 1 4 .

(Dover reprint)

21. Integral Equations FRANK , P. and VQN MISES , R., Differential- und Integralgleichungen der Mechanik und

Physik, M. S. Rosenberg, New York, 1 9 4 3 . (Dover reprint)


Collections of formulas ADAMS, E . P. and HIPPISLEY , R. L., Smithsonian Mathematical Formulae and Tables

of Elliptic Functions, Smithsonian Institution, Washington, 1922. JAHNKE, E . and EMDE , F., Funktionentafeln mit Formeln und Kurven, B. G. Teubner,

Leipzig, 1938. (Dover reprint) MADELUNG, E. , Die mathematischen Hilfsmittel des Physikers, Julius Springer, Berlin,

1936.

Chapter 2

S T A T I S T I C S

BY JOSEPH M . CAMERON Statistical Engineering Laboratory

National Bureau of Standards

This chapter on statistics was added, at the special request of numerous physicists, to elaborate some of the formulas of mathematical statistics as given in abbreviated form in Chapter 1. Experimental physicists have become increasingly aware of the importance of assigning to their measures some parameter such as mean error, probable error, or limit of error to represent the precision of their measurements and to serve as a yardstick for judging differences. Nevertheless, the literature of physics abounds with examples where the scientist has failed to take full advantage of modern statistical methods. This chapter represents an attempt to give the major formulas of mathematical statistics in a form directly useful for the research student.

1. Introduction

1.1. Characteristics of a measurement process. A sequence of measurements on the same subject derived from the same measurement process (i.e. with the environmental conditions and procedures maintained throughout the sequence) tends to cluster about some central value for the sequence. If, as the number of measurements is increased indefinitely, the average for the sequence approaches, in the probability sense^ a value /x, then fju is the limiting mean of the sequence.

An individual value xt differs from the limiting mean fju because of uncontrolled or random fluctuations in the environmental conditions or procedures. These individual values will follow some frequency distribution about the limiting mean. The specification of this distribution is made in terms of a mathematical frequency law involving several parameters including the limiting mean /z, a measure of the dispersion about the limiting mean, and other parameters as necessary.

The dispersion of values about the mean is measured in terms of the square root of the second moment about the mean, called the standard devia-

107

108 STATISTICS §1-2

tion, a. This standard deviation is the limit, in the probability sense, of the square root of the average value of the squared deviation of the individual values from the mean as the number of values is increased indefinitely.

Formulas for estimating the parameters of a distribution from a finite set of data, and methods of making statistical tests of hypothesis concerning these parameters are presented in this chapter.

1.2* Statistical estimation* The experimenter has at hand only a finite set of measurements which are considered a subset of an infinite sequence. He will, therefore, be in ignorance of the parameters of the process such as the limiting mean /x, the standard deviation cr, and any other parameters that may exist. The finite set of data is to be used to estimate the unknown parameters. For example, the average or the median may be used to estimate the limiting mean. The function of the data used for estimation is called an estimator; the numerical value obtained by using this estimator on a set of data is called an estimate.

Estimators of parameters may be either point estimators, such as median or average, or may be interval estimators, such as confidence limits.

A sequence of estimates, generated by use of an estimator on small sets of values randomly selected from some distribution, will have a limiting mean, a standard deviation, and follow some distribution. Just as one technique of measurement may show superiority over another, so also may one method of estimation (i.e., one estimator) show superiority over another in terms of bias and precision.

The exact form of the underlying distribution of the basic measurements cannot be settled, and upon the appropriate choice of the form of the distribution depends the correctness of the inferences drawn from the data.

1.3. Notation. Unless otherwise stated in the formulas that follow : Individual measurements are indicated by

\n M y1 \* V

1 2 3 • • • % • • • TI

where the subscript refers to the order in which the measurements were made.

Each xi is assumed to be independent in the statistical sense from all other measurements in the set, and all xiare assumed to follow the same distribution.

Measurements ranked in order of magnitude are indicated by x(l)> x(2)i ••• ••• x(n)

such that x(1) < #(2) < ... < xn)

§ 2 . 1 STATISTICS 109

Parameters of distributions are indicated by Greek letters, /x, cr, /?, etc., while estimates of these parameters are indicated by italic letters x, s> b, etc.

2. Standard Distributions

2.1. The normal distribution.* A variate that follows the frequency function

f(x) = 7 = e-^-ô* < x < m (!)

is said to be normally distributed with

mean value of x — \L variance of x = cr2

The normal distribution is quite often a fairly good approximation to the distribution of random fluctuations in physical phenomena. This general applicability of the normal distribution is verified by experience and supported in part by theory.

By virtue of the central limit theorem, which states that for certain conditions the sum of a large number of independent random variables is asymptotically normally distributed, the distribution of the random fluctuations in a measurement process tends toward normality because the variations are due to a multitude of minor random variations in the process.f

2.2. Additive property. If xlt x2i xn are normally and independently distributed with means / x 1 / x 2 , i^ni and variances c r ^ , c r 2

2 , c r n2 , then

C1X1 ~t~ C2X2 "f" ••• ~h CnXn

where the ci are arbitrary constants and will be normally distributed with

n n mean ^ c^ and variance ^ câr?

Tables of the standardized normal distribution

V 2 7 7 J-X

to 15 decimals are given in Tables of Normal Probability Functions, National Bureau of Standards, Applied Mathematics Series, No. 2 3 , 1 9 5 3 .

t CRAMER , H., Mathematical Methods of Statistics, Princeton University Press, 1 9 4 6 , pp. 2 1 3 - 2 2 0 ; and Random Variables and Probability Distributions, Cambridge Tracts in Mathematics, No. 3 6 , 1 9 3 7 , p. 1 1 3 .

1 1 0 STATISTICS §2 .3

2.3. The logarithmic-normal distribution. If y = In x is normally distributed with mean /x, and standard deviation <r, then x will have mean £

and variance ax2.

am* = &+°\e°%-l) = I) ( 2 )

Note : The use of the average of the x's as an estimate of £ is inefficient when the variance of the y's is greater than 0 . 7 . The use of s2 as an estimate of ax

2 is inefficient when the variance of the y's is greater than 0 . 1 . For efficient estimators see FINNEY, D. J., Supplement to Journal of the Royal Statistical Society, Vol. 7, No. 2 , 1 9 4 1 , pp. 1 5 5 - 1 6 1 .

2.4. Rectangular distribution. If the probability that x lies in the interval x0 + dx is the same for all values of xQ and x is restricted to the interval (a, /?), then x follows the rectangular distribution with frequency function

(«<x<P) ( 1 )

mean value or x = —^—, variance or x = -—

2.5. TheX2 distribution. The sum of squares of n independent normal variates having mean zero and variance unity follow the X2 distribution for n degrees of freedom. The X2 distribution enters in statistical tests of goodness of fit, homogeneity, and a variety of other purposes because the standardized variable

estimate of mean — mean standard deviation of estimate of mean

tends to be asymptotically normal with mean zero and variance unity. T h e ^ 2 distribution for n degrees of freedom has frequency function

/ ( I 2 ) = 2 w 2 r ( w / 2 ) * - * A W - 2 ) / 2 ( o < r < - ) ( i )

mean o f^ 2 = n, variance o f^ 2 =• 2n

The X2 distribution has the additive property that the sum of k independent X2

variates based on nly n2, n3, nk degrees of freedom respectively follows a X2 distribution for

k

i=l degrees of freedom.


A table of values of X2 exceeded with probability P is given in most texts on statistics.

2 .6 . Student's ^-distribution. In sets of size (n + 1) from a normal distribution the ratio of the average to standard deviation (t = xjs) follows Student's distribution with n degrees of freedom. Frequency function for the /-distribution is

, / x r [ ( » + l ) / 2 ] 1 1 , x ,ix

ti mean value of t = 0 variance of t = = (n > 2)

n — 2

Tables of the values of £ exceeded in absolute value with probability a are given in most modern texts on statistics, i.e., t0 for which

2 . 7 . The F distribution. If s^ and s22 based on m and TZ degrees of

freedom, respectively, are two independent estimates of the same cr2 for a normal distribution, F = s1

2/s22 will follow the F distribution for (m,ri)

degrees of freedom (the number of degrees of freedom for the numerator is always quoted first).

The frequency function of F is

( F , r [ ( m + ») /2] F - » > / » m^p^^ m ' r ( » / 2 ) r ( » / 2 ) \ « / [1 + (w/n)F]<™+"»/2 l U ^ ^ ; l U

mean of F = j? (n > 2) w — 2 7

variance of JF = 2(w + w - 2 ) m(w — 4)

( n > 4 )

Tables of values of F exceeded with probability 0.05 and 0.01 are given in G . W . SNEDECOR, Statistical Methods, Iowa University Press, 4th edition (1946), and in a number of recent texts on statistics.

2 . 8 . Binomial distribution. If on any single trial, the probability of occurrence of an event is P and the probability of nonoccurrence of the event is 1 — P, the probability of k — 0, 1, n occurrences in n trials is given by the successive terms in the expansion of

[(l-P) + P]n = (l-P)n + n(l-P)n-1P+... (1)

112 STATISTICS §2.9

The average number of occurrences in n trials will be

nP

and the standard deviation of the number of occurrences will be

VnP(l - P)

Tables of individual terms and cumulative terms for P = .01 (01).50 and n = 1(1)49 are given in Tables of the Binomial Probability Distribution, National Bureau of Standards Applied Mathematics Series, No. 6, 1950.

2.9. Poisson distribution. If the probability of the occurrence of an event in an interval (of time, space, etc ) of length dx is proportional to the length of the interval, i.e., probability of occurrence is equal to, say, A dx, the probability of k independent occurrences in an interval of length x is given by (for fi = Xx)

The average number of occurrences is JLC ; the standard deviation of the number of occurrences is <y//z. Additive property : If klf k2, kn independently follow Poisson distributions with means [ily JJL2, \in, then

n

X*, i=l

n also follows a Poisson distribution, with mean ^ /x2-.

Note : For P small (less than 0.1) the Poisson may be used as an approximation to the binomial taking JJL = nP.

Tables of individual terms and cumulative sums are given in MOLINA, E. C , Poissons' Exponential Binomial Limit, D . Van Nostrand Company, Inc., 1947.

3. Estimators of the Limiting Mean

3.1. The average or arithmetic mean. Symbols .

1 n

x = y. X n n i=l

If the x.L have limiting mean /x and standard deviation cr,

limiting mean of x = fi

s.d. of x — aj\/n


3.2. The weighted average. If each xi has assigned to it a weight of wiy

the weighted mean is defined as n t

J^L (1)

If the xi have same limiting mean /x and standard deviation cr,

limiting mean of weighted average = \L

A " s.d. of weighted mean = a — —

3.3. The median. The median is that value that divides the set of values xH) into two equal halves. The median of n values is

*(n+i>/2 for » 'odd

i(*W2) + *(n/2+u) for w even

If the xi have limiting mean /x and standard deviation a and if the distribution of the #'s is symmetrical about /x,

limiting mean of the median = \L

If the xt follow a normal distribution the standard deviation of the median approaches, for large w,

J ^ - - ^ = 1.2533 ^ (1)

For values of n < 10 the s.d. is approximately 1.2cr/-y/w.

4. Measures of Dispersion

4.1. The Standard deviation. Symbol s.

114 STATISTICS §4 .1

If the x follow a normal distribution with limiting mean JJL and standard deviation cx,

limiting mean of s — T(n/2) r t ( » - i ) / 2 ] V n - 1

1

(2)

= ao ~ \\ 1 2(n - 1)

s . d . o f , = V l - « 2 a ~ ^ = = = (3)

The distribution of s is approximately normal for n > 30.

Note: In all statistical analysis involving comparing of averages or precision of two processes, the square of the standard deviation, or variance, is used. If such analysis is intended, the estimator s is to be preferred over other estimators of standard deviation because s2 has a limiting mean cr2, the other estimators do not. These alternate estimates are :

a. The estimator

V n v n for which, if the ^ fo l l ow a normal distribution,

X(*'-*)2 In-I fil. = * l n — ± s (4)

. . . . r In - 1 In-I limiting mean of <\J —-— s = / y -—-— do

s.d. of \ - --s = A / 2 - A / I

\ n y n b. The estimator

1 T[(n 1 ) 2 ] In-I . AS (*' ^ a * ~~ r(n/2) 'V 2 ' \ " n -T

(5)

for which, if the x follow a normal distribution,

limiting mean of — s = a

§4.2 STATISTICS 115

c. The estimator

0.67449 k 1 M

the so-called " probable error." For the case of a normal distribution has mean value

0.67449 J - l-aa

and standard deviation

0.67449 ^n-n-^ Vl~a2a

For a normal distribution with mean JJL and standard deviation cr , the interval (/x — 0.67449 cr , /x + 0.67449 c r ) will include 50 per cent of the frequency distribution. When cr is estimated from finite sets of data such an exact probability statement cannot be made (see tolerance limits).

4.2. The variance s2 = £ (x% ~~ x)2l(n — 1)- Whatever the distribution i=l

of #'s, the limiting mean of s2 = c r 2 .

If the x's follow a normal distribution with standard deviation cr,

l~2~ standard deviation of s2 = \ T c r 2 (1) \ n — 1

and (n — \)s2ja2 follows a % 2 distribution with (n — 1) degrees of freedom.

1 4.3. Average deviation. — £ , \xi ~ x I

n S If the xi are normally distributed, limiting mean /x and standard deviation cr ,

limiting mean of average deviation = \ j — ^~~n~~CJ (^)

A t A • ,2Vn-l s.d. of average deviation = 77 n

— + Vn(n — 2) — n

(2)


for large n. Note that in comparing two average deviations, account must be taken of any difference in the number of observations in the two estimates.

4.4. Range: difference between largest and smallest value in a set of observations. xn) = x(1). If the x\ follow a normal distribution with limiting mean JJL and standard deviation cr,

limiting mean of range = (constant) cr

Note : Tables of this constant for different values of n are given in Biometrika, Vol. 24, p. 416, along with values of the standard deviation of the range. With the use of these tables an estimate of the standard deviation can be obtained by dividing the range of the data by this constant. For example : for sets of 5 measurements the constant is 2.33,

10 measurements the constant is 3.08, 20 measurements the constant is 3.73, 30 measurements the constant is 4.09,

100 measurements the constant is 5.02.

5. The Fitting of Straight Lines

5.1. Introduction. When the mathematical relationship or law between two variates is known or assumed to be linear, and exact knowledge of the relationship is obscured by random errors in the measurement of one or both of the variates, the formulas of this section are applicable.

A clear distinction must be made between the case of estimating the parameters of a linear physical law and the case of estimating the parameters of linear regression. In the physical law case there are two mathematical variables, X and Y, related by a linear equation. Our inability to measure without error obscures this relationship. In the linear regression case the average values of y for given x's are related to x by a linear equation. In the linear regression case there is a bivariate distribution of x and y. There are two distinct regression lines (that of y on x and x on y) and there is a correlation between x and y. The regression lines and the correlation are properties of the underlying bivariate distribution.

5.2. The case of the underlying physical law. Let X and Y represent the variables of the physical law, and :

a. The two variables X and Y are related by a law of the form Y = oc + fiX or its converse X = (—a//?) + (VP)Y.


b. At any X repeated measurements y, of the variable Y, are dispersed about Yt = OL + /3Xi with a standard deviation to be called oy , the functional form of the error distribution being the same at all Xit

c. At any Yi repeated measurements, xiy of the variate X are dispersed about Xt = —oc/ft + (ljp)Yi with a standard deviation to be called aXi, the functional form of the distribution of errors being the same at all Yit

d. In practice there are obtained n paired values xiy^ which correspond to paired values (XiyY) from the linear law. The problem is to obtain the best estimate, Y = a + bX of the linear law Y = oc + /3X from the n values

If one variable, X, is known to be measured without error, the following tables (pages 118-119) are applicable.

If both variates are subject to error, i.e., if cr = ay and aXm = orx and the error distribution is normal, the following technique is applicable.

Step 1. Arrange the n paired values (xiyy^ in order of magnitude according to one of the variates, say x.

Step 2. Divide this ranked set of n values into three nearly equal groups, so that in the two extreme groups there are the same number, say k. Thus the center has n—2k.

Step 3. Compute the averages, xx and yx of x's and y's in the first group, the averages x3 and yz of the #'s and y's in the third group, and the grand averages x and y.

Estimator of slope b' = y*-yi X g X j

Estimator of intercept

Estimator of error in j ' s ,

a = y — b'x

n n ~|

yZ(yi-y)2--b'%(Xi-x)(yi--y) / - i Estimator of error in JC'S,

The distributions of the estimators depend on the value of the unknown slope j8. See WALD, A . , Annals of Mathematical Statistics, Vol. 11, 1940, p. 284; BARTLETT, M. S., Biometrics, Vol. 5, 1949, p. 207.

120 STATISTICS §6 .1

Note: One of the estimates, sy2 or sx

2y can be negative. Such negative

values are to be expected more often when n is small, or when the parameter (crx

2 or Gy2) being estimated is near zero. The distribution of the estimates

has not been worked out.

6. Linear Regression

6.1. Linear regression. If two variates x and y vary concurrently (as for example, the height and weight of men age 30) and a linear relation exists between the average value of y and the corresponding value of x, and also a linear relation between the average value of x and the corresponding y, the formulas of this section are applicable.

Parameter Estimator

a — y — bx

n S (x€— x) (yi — y)

b = —

n 2 t(x€ — x)2

i=i n S Xi — ofyiyi — y)

V n n

i=i i=i a = x — by

S Xi — x)(yi — y)

1 n — 1 i=

1 n — 1 i=

2 ( y t - y ?

Varia

1 , ( 1 - P 2 )

— ( 1 - P 2 ) 2

for large /z

n— 3 av

( i - p 2 )

§ 7.1 STATISTICS 121

For a given xit the y's have a variance ay2, and mean value

yi-=oc + flXi

and for a given y, the x's have a variance era, 2 , and mean value

xt = OL + p'yt

where y and x are the averages of an infinite number of values at y and x, respectively. Thus there are two lines to determine, and further there is the parameter p, the correlation between x and j>.

7. The Fitting of Polynomials

7.1. Unequal intervals between the x's. Let n paired values (xity^ be observed. If the variate x is measured without error and the variate y is related to x by the equation

y = a0 +cx^ + OL2X2 + ocsxz + ... (1)

and the errors in y follow a distribution with mean zero and standard deviation a, the constants are estimated by minimizing the sum of squares of the y deviations, i.e., the estimates aQy aly a2i ... of a 0 , ocly oc2, ... are those values for which

n

(yi — a0 — axx — a2x2 — . . . ) 2 = minimum (2)

This minimization process leads to a set of simultaneous equations :

Hy — naQ — a-JLx — a2Hx2 — ... = 0 \

Tixy — a0TiX — a-^Lx2 — a^x3 — ... = 0 > (3) Hx2y — a^x2 — a-^Lx* — a2Hx* — ... = 0 etc. )

where the summations are over i = 1, 2, n. For the estimators of the parameters of linear equation y = oc0 + cq* the

solution is

I Dry 2a?2 I I Hx IZxy I (4) a 0 = — ^ fll = _ _

where D2= n Ex j | HiX ''JLJX2 I

and the sum of squares of deviation of the y s from the line is

S22 — Hy2 — a^Ly — a-^Lxy.


For the estimators of the parameters of a parabola, y = oc0 + ocxx + &2x2, Ey Hx Hx2

Hxy 2 # 2 2 # 3

1iX2y S # 3 2 # 4

n 2 y Hx2

Hx Hxy 2 # 3

%x2 %x2y

do =

I n TiX 2 y YiX E* 2 ILxy

I Hx2 TiXs YiX2y

(5)

where D3 =

A , n Ex E# 2

S# 21A;2 E# 3

E# 2 E# 3 E# 4

and the sum of squares of y deviation from the parabola is

Ss2 = Ey 2 — a0Hy — a-^Lxy — a2lLx2y (6)

The extension to polynomials of higher order is obvious. The estimate of the standard deviation of the y determination

and s =

n — 2 S 3 2

n — k

for the line

for the parabola

for a polynomial of degree k — \

(7)

(8)

(9)

For testing the significance of any of the constants fitted, the reduction in the total sum of squares of the y's is studied as follows :

Degrees of Mean freedom square

Total sum of squares . . E (vt-— y)2 = *S"i2

i=l Deviation from linear S%2

Reduction of sum of squares due to linear terms — £ 2

:

Deviations from quadratic S32

Reduction of sum of squares due to quadratic terms s2

2—*sy etc.

n— 1

n — 2

1 n—3

i S22l(n-2)

j SMn-3)

j I So" Sg2


T o test the significance of a coefficient, ak, compute

- s, 2

If the distribution of errors in the y measurement is normal, F will follow the F distribution for 1, n — k degrees of freedom. If the computed F is greater than the value of F for 1, n — k degrees of freedom exceeded with probability a, the reduction in sum of squares due to fitting the constant a1c

(coefficient of x^1) is regarded as significant at the 100 a: per cent level of significance.

Each of the ak are seen to be a linear function of them's, so that the standard deviation of any particular at can be obtained by writing

and using the formula for the variance of a linear function, noting that the mean value of yi = a 0 + + oc2x^ + • • • •

7.2. The case of equal intervals between the x's — the method of orthogonal polynomials. In order to find the best fitting polynomial

y = a0 + axx + a2x2 + ... (1)

when the x values are evenly spaced, it is convenient to fit instead the polynomial

y = A + Bg1' + C£t' + ... , ( 2 )

where £ 2 ' , '> ••• a r e orthogonal polynomials of degree 1, 2, 3, ...

£1' = (x — x)X,

£2' (x - xf 12—

t >t > r\n2-r2) ,

The A's are chosen so that all the coefficients of these polynomials are integers. The coefficients of these orthogonal polynomials and a description of the method of analysis are given in FISHER, R. A., and YATES, F. , Statistical Tables, 4th edition, Hafner Publishing Company, New York, 1953 for £ / , to f B ' , for n = 3(1)75.


7.3. Fitting the coefficients of a function of several variables. Just as a distinction was made in the case of the straight line between the underlying physical law situation and the linear regression situation, so also must this distinction be preserved with the extension to this multivariate case.

If Y is related by a physical law to variates Xlt X2, •.. by the equation

y = . • + OL2X2 + ... (1)

one is interested only in estimating the coefficients ock. There is no question of the correlation between, say, Xx and X3. In the physical law case when Xlf X2, and XS are regarded as free from error and Y as subject to errors following a distribution with mean zero and standard deviation cr, for n values (yit Xlif X2I, ...) the estimators ak for the parameter ock are, for the case of the law Y = oc0 + oc1X1 + OL2X2,

Ey S Z 2

HXtf D Z j 2 1LXXX2

Y,X2y ZX1X2 Y*X2

n Ey E X 2 | IIX1 Y*XIY S I A ; E X 2 E X 2 y E X 2

2 i

(2) n IIX1

E X X SXj . 2

E X 2 ÛX^X 2

where « S X X

E X 2 HX±X2

E X 2

YIXXX2

The sum of squares of deviations of j> from the fitted equation is

S 32 = Ey 2 — a0Ey — aJLXiy — a 2 EX 2 j / - (3)

The extension to include cases of more variables is straightforward, and the tests of significance of the coefficients is analogous to the case of fitting a polynomial.

7.4. Multiple regression. In the multiple regression case where a number of variates are varying concurrently (as, for example, weight of males age 30, as a function of height, wrist circumference, and waist girth) is treated in R. A . FISHER, Statistical Methods for Research Workers, Oliver and Boyd, Section 29. In multiple regression there are partial correlations and partial regression coefficients to be estimated.


s . d . o f j > = ^ 1 P \ (2)

100(1 —oc) per cent confidence intervals PlyP2) for the parameter P of the binomial for k occurrences in a sample of n are given by solving

*(i -py-ôLJi (3)

for P, giving the lower limit Ply and

2 ) ( " W - i » ) « - ' = «/2 (4)

for P, giving the upper limit P2. The computation is simplified by using the incomplete beta function (see

MOOD, A. M., Introduction to Theory of Statistics, McGraw-Hill Book Company, Inc., 1950, pp. 233-235).

8.2. Estimator of parameter of Poisson distribution. If in an interval of x units, k occurrences are observed, the estimator for the parameter A of the Poisson is k/x, which has a standard deviation of \/Xjx.

If n independent estimates of A, kjx^ k2/xn, kjxn are available, the best estimate of A is

n

n

i=l

which has standard deviation . X i '

8. Enumcrativc Statistics

8.1. Estimator of parameter of binomial distribution. If in n trials, m occurrences of some event are observed, then the estimator for the parameter P is

P = - - ( D

This estimate will have a standard deviation

1 2 6 STATISTICS § 8 . 3

The average of the ratios has standard deviation

To test the conformance of a series of n values of kt to the Poisson law where the interval length is the same for all determinations, compute

n

X 2 = i — - k — ( 5 )

where k = (I In) X fy. i=i

This statistic will have approximately the X2 distribution for n — 1 degrees of freedom.

8.3. Rank correlation coefficient. N objects are ranked by two methods. Denote the ranking of the zth object by the first method by ru and its ranking by the second method by r2i, and the difference di = rxi — r2i.

The rank correlation coefficient is defined as

i=l

1 ~N(N2-l)

For a complete treatment of rank correlation see KENDALL, M . G., Rank Correlation Methods, Charles Griffin & Co., Ltd., London, 1 9 4 8 .

9. Interval Estimation

9.1. Confidence interval for parameters. Instead of a point estimator of a parameter, an interval estimator may be used. A random interval (L 1 ,L 2 ) , depending only on the observed set of data having the property that 100(1 — lot) per cent of such intervals computed will include the value of the parameter being estimated, is called the 100(1 — lot) per cent confidence interval.

No probability statement can be made regarding an individual confidence interval since a particular interval either includes the parameter, or it does


not. It is the method of estimation that carries with it the probability of correctly bracketing the parameter.

For a given distribution of measurements it is possible to set down the distribution of estimates of a parameter derived by using a given estimator and a sequence of random sets from the original distribution. For example, the probability that x based on n measurements lies in the interval (fju — 1.96o-/Vw, n + 1.96'<rl<\/n) is 0.95. From this it follows that the random interval (x — 1.96 cr/y7n, x + 1-96 a/^/n) will include the parameter JJL with probability 0.95, and the interval (x — 1.96 o/y/w, x + 1.96 o/y'w) is therefore the 95 per cent confidence interval estimator of /x.

9.2. Confidence interval for the mean of a normal distribution. For a set of n measurements from a normal population the interval (LlyL2) where

Lx = X — * ( 2 a > n - l ) (1)

and L 2 = x + f ( 2 a ,n-i) 7 7 ^ ( 2 )

where / ( 2 a n _ 1 ) is the value of Student's t for n — 1 degrees of freedom exceeded in absolute value with probability 2a, defines the 100(1 — 2a) per cent confidence interval for JJU, the mean of a normal distribution.

9.3. Confidence interval for the standard deviation of a normal distribution. From a set of n measurement (i.e., an estimate of standard deviation based on n — 1 degrees of freedom) from a normal distribution the interval (L l 5 L 2 ) where

A ( « ,n— 1)

and L t = J ( 2 ) * / I ( l -a ,n - l )

and where X2(OL n-i) l s t n e v a h i e of ^ 2 for ^ = w — 1 exceeded with probability a, defines the 100(1 — 2a) per cent confidence interval for cr.

Note: (Lj 2 , !^ 2 ) defines the 100(1—2a) per cent confidence interval for cr2.


9.4. Confidence interval for slope of straight line. If the variate x is free from errors and the variate yiy

= ( « + £ * , ) + € , . ( 1 )

where e is a random variable, normally distributed with mean 0 and standard deviation a for all iy then (LlyL2) is the 100(1 —2a) per cent confidence interval for /?, the slope of the line.

L1 = b — £(2a,w-2)^b (2)

L2 = b -\- £(2a,w-2)^b (3)

where t2<x,n-2) 1S t n e value of Student's ty exceeded in absolute value with probability 2a.

Note : If the intercept is known to be zero, replace t2txn-2) by ^ 2 a ,w-i) m

the formulas for Lx and L2.

9.5. Confidence interval for intercept of a straight line. Under the same conditions required for a confidence interval for the slope,

L1 = a t(2a,n-2)sa U)

^ 2 — a ~f" ^(2a ,n -2 )â (2)

define the 100(1 — 2a) per cent confidence interval estimator for a, the intercept of a straight line.

9.6. Tolerance limits. Limits between which a given percentage, P, of the values of a distribution lie are called tolerance limits. For example, for the normal distribution with mean /x and standard deviation cr, 95 per cent of the values lie between the limits \x — 1.96cr and \L + 1.96cr. If x and s are estimates of /x and cr, it cannot be stated that the limits x — 1.96s and x + 1.96 include 95 per cent of the population. Here x + Ks (where K is a constant) is a random variable which is approximately normally distributed with mean /x + Ks and approximate standard deviation cr-\/ l/n -f- K2j2n — 1) where n is the number of observations upon which x is based and (n — 1) is the number of degrees of freedom in the estimate s.

The value of K can be adjusted so that the limits (x — Ks) and (x — ks) will in a given percentage y of the cases include a given proportion P of the original distribution. See Techniques of Statistical Analysis, edited by C. EISENHART, M. W . HASTAY, and W . A . WALLIS, McGraw-Hill Book Company, Inc., 1947, Chapter 2, for tables of values of K for various n, y, and P.


10. Statistical Tests of Hypothesis

10.1. Introduction. Investigations are concerned not only with estimating certain parameters, but also with comparing the estimates obtained with certain assumed values or with other estimates. For example, an average value for a certain quantity is derived from two separate experiments. The question is asked : Can the averages be considered in agreement ? In this case the hypothesis to be tested, called the null hypothesis, is that the limiting means of the distribution from which the averages arose are identical.

The specification of the statistical test requires a statement of the alternate hypothesis against which the null hypothesis is tested. The statistical test will lead to either acceptance or rejection of the null hypothesis.

Any statistical test may result in a wrong decision. Errors of two kinds may be committed : Type I error: rejecting the null hypothesis when it is in fact true and, Type II error: accepting the null hypothesis when it is in fact false. For any statistical test the risk a of making a Type I error can be made as small as desired.

A statistic T, some function of the observed value, is chosen in the light of the null and alternate hypotheses. When the null hypothesis is true this statistic will tend to fall within a certain range; when the alternate hypothesis is true the statistic T will tend to fall in some other range. These ranges will in general overlap. The value selected to serve as the boundary between these ranges determines the risks of making a wrong decision.

The null hypothesis is rejected if an observed value of a test statistic exceeds the critical value. In this section the critical value specified is such that there is a chance cx of wrongly rejecting the null hypothesis. The test is said to be conducted at the 100a per cent level of significance.

If in conducting a test of a hypothesis at the 100a per cent level, a value is obtained that falls short of the critical value, the statement can be made that " there is no evidence of a difference." If the observed value of a test statistic exceeds the critical value, the statement can be made that " there is evidence of a difference since as divergent a set of values as found would occur by chance only 100a per cent of the time if no real difference existed."

10.2. Test of whether the mean of a normal distribution is greater than a specified value.

a. Observed values: x and s based on n measurements considered as a set of values from a normal distribution with mean /x, and standard deviation cr .


b. Null hypothesis:

c. Alternate hypothesis:

/x = u . 0

/x > / x 0

d*4 Test statistic : x sl\/n~

t =

£ follows Student's 2-distribution for w — 1 degrees of freedom if null hypothesis is true.

e. Rejection criterion: Reject the null hypothesis if the observed value of t is greater than the value of t for n — 1 degrees of freedom, exceeded with probability oc.

10.3. Test of whether the mean of a normal distribution is different from some specified value.

a. Observed values: x and s based on n measurements considered as a set of values from a normal distribution with mean / x , and standard deviation cr.

b. Null hypothesis : fx = /x0

c. Alternate hypothesis JJL ^ /x0

d. Test statistic : t = ^ °° ~ f^l]

sjy n t follows Student's ^-distribution for n — 1 degrees of freedom if null hypothesis is true.

e. Rejection criterion : Reject the null hypothesis if the observed value of t is greater than the value of t for n — 1 degrees of freedom, exceeded in absolute value with probability oc. (This is equivalent to the value of t exceeded with probability a/2).

10.4. Test of whether the mean of one normal distribution is greater than the mean of another normal distribution.

a. Observed values: x± and s± based on n measurements considered as a set of values from a normal distribution with mean and standard deviation cr, and x2 and s2 based on m measurements considered as a set of values from a normal distribution with mean / x 2 and standard deviation cr.

b. Null hypothesis : = fju2

c. Alternate hypothesis : > / x 2

§ 1 0 . 5 STATISTICS 131

d. Test statistic

t = y/(m + n)jmn V[(n — l)*i2 + (m — ^)s2,]l(m + n — 2)

t follows Student's ^-distribution for (n + m — 2) degrees of freedom when null hypothesis is true.

e. Rejection criterion: Reject the null hypothesis if the observed value of t is greater than the value of t for (n + m — 2) degrees of freedom exceeded with probability a.

10.5. Test of whether the means of two normal distributions differ.

a. Observed values : xx and s1 based on n measurements considered as a set of values from a normal distribution with mean JJL^ and standard deviation cr , and x2 and s2 based on m measurements considered as a set of values from a normal distribution with mean /x2 and standard deviation cr.

b. Null hypothesis : = /x2

c. Alternate hypothesis : /xx /x2

d. Test statistic : I x, — xn\

t V(m + n)/n \/[(n - l)s* + (m - 1>22]

t follows Student's ^-distribution for (m + n — 2) degrees of freedom when the null hypothesis is true.

e. Rejection criterion : Reject the null hypothesis if the observed value of t is larger than the value of t for (m + n — 2) degrees of freedom exceeded in absolute value with probability oc.

Note : For the case where the distributions do not have the same standard deviation, see FISHER, R. A . and YATES, F., Statistical Tables, 4th edition, Hafner Publishing Company, New York, 1953, pp. 3-4.

10.6. Tests concerning the parameters of a linear law. In the case where X is free from error and the errors in the observed values of y follow a normal distribution with mean zero and standard deviation cr, the statistics

t = -=&L (1)

and t = - - - ^ (2)


follow Student's ^-distribution for degrees of freedom equal to the number of degrees of freedom in the estimate of sb and sa, (which will be n — 2 for both when the line does not pass through the origin, and n — 1 for sb when it is assumed a priori that oc = 0) when the null hypothesis = j30 and oc = ocQ

are true. The tests of hypothesis concerning j8 can therefore be made in the same

manner as tests regarding the mean of a normal distribution.

10.7. Test of the homogeneity of a set of variances. Let s2\ sk be k variance estimates based on nv n2, ...>nk observations respectively. Then the statistic

2 3 0 2 5 9 \N log S* - X (»< ~ 1) l o § (1)

where N:

S*.

X ( » < - i )

X ( n , - 1 K 2

i=l N k

S t v ^ - i ) ] - ! / ^ c = 1 + ^ X F - r i )

has approximately the X2 distribution for k — 1 degrees of freedom under the null hypothesis that the s

2 are all estimates of a common cr2. The null hypothesis is rejected if the observed value of B is greater than the value of X2 f ° r ( ~ 1) degrees of freedom exceeded with probability oc.

10.8. Test of homogeneity of a set of averages. If xv x2> xn

are n averages, each based on k measurements each, the statistic

F

where

k^x-xf\ / ( n - 1 ) <=i J

=i

1 n

X = 7, Xi n M

(1)

i=l j=l


follows the F distribution for (n — 1), and n(k — 1) degrees of freedom under the null hypothesis that limiting mean of xt is the same for all i. Reject the null hypothesis if the observed value of F is greater than the value of F for n — 1, n(k — 1) degrees of freedom exceeded with probability a.

10.9. Test of whether a correlation coefficient is different from zero. If r is the observed correlation coefficient found in the linear regression case, assumed to be an estimate of p of a bivariate normal distribution, then in order to test the null hypothesis that p = 0 against the alternative hypothesis that p ^ 0 compute the statistic

1 = ! v T ^ 2 (i) When the null hypothesis is true £ follows Student's ^-distribution for (n — 2) degrees of freedom. Reject the null hypothesis if the observed value of t is greater than the value of Student's ^-distribution exceeded in absolute value with probability a.

10.10. Test of whether the correlation coefficient p is equal to a specified value. Let

1 , 1 + r # = —- log

2 5 1 — r ( 1 )

r 1 i 1

£ = l 0 g : 2 & 1 - p

where r is the observed correlation found in the linear regression case and assumed to be an estimate of p of a bivariate normal distribution, then (z — I) is approximately normally distributed with mean zero and standard deviation approximately ^\\n — 3).

Thus a test for the significance of the difference of r from any arbitrary p is given by computing the statistic

; » ( £ H - ( £ g ) Under the null hypothesis that p = p 0 , t will be approximately normally distributed with mean zero and standard deviation a = 1.

Reject the null hypothesis if t is greater than the standardized normal deviate exceeded with probability a/2 when the alternate hypothesis is that p p0.


11. Analysis of Variance

11.1. Analysis of Variance . Consider a two-way classification of data as shown in the table.

1 2 Column

j c Row

averages

1 X12 xij xic xRl

2 X21 XR2

Row i xn xij xRi

r xri Xrj xrc xRr

column averages xCl XC2 xCo xCc X = grand average

It is a property of numbers that

r c

XX(*«~~*)a

i=l j=l

can be broken up into three components as follows :

c (Deviation of row r

averages about grand average) = c X (%Ri — #) 2 (1) i=l

r (Deviation of column c

averages about grand average) ---= r X (%C3 ~~ %)2 (2)

3=1

r c

Residue = X X (Xi* ~ * R i ~~ * C j + ^ It is assumed in what follows that the xi3- are normally independently distributed with variance cr2, about means fjLi3-

Pa = Hô + &m + 8 C j ( 3 )

where /x0, is the grand mean, SRi and 8Cj are the deviations of the row and column means, respectively, from the grand mean. Two models of analysis of variance are possible.

§ 11.1 STATISTICS 135

a. Model I. The parameters oRi and oCJ are fixed constants.

For example, if the rows represent different machines and the columns different operators then the oRi are the deviations of the individual machines performance from the grand mean, and oCJ are the operator biases.

The row averages and column averages are estimators of jjlq + 8^- and /x0 + oCj respectively.

In order to test the hypothesis that the row means are all identical or that the column means are all identical, the following computations are made :

! Degrees I Sum of squares i of | Mean square

: j j freedom j

deviation of row averages . . .

Mean square is

estimate of

R = c 2 (xRi — x)2 r — 1 R

r— 1

deviation of column averages . j C = r (xCj — x)2 j c — 1

3=1 i

lesidue

Total .

T — R—C \ (r— 1 ) (c— 1 )

rc — 1

C

c — T

T—R — C

i=i

r—\

3=1

T o test the null hypothesis that the row means are identical, that is, that

X 8*«2 = 0

i=l

compute the statistic

F = fci> (4) ( R _ . R _ 0 / [ ( r - l ) ( C - l ) ] >

Under the null hypothesis F follows the F distribution for r— 1, (r — 1) (c — 1) degrees of freedom. A similar test can be made for the column means.

136 STATISTICS §11-1

b. Model II. The parameters are components of variance.

For example, if the rows represent different samples of the same material and the columns represent different days on which the tests were conducted, then S^- and SCj are random variables. These random variables will have mean zero and variances orR

2 and c r c2 , respectively, and in that which follows

are assumed to be normally distributed. The computations are the same as shown in Model I.

Sum Degrees Mean of of Mean square square is

squares freedom estimate of

Deviation of row averages . . . .

R r— 1 R

r— 1 a2 + caR

2

Deviation of column averages . . . .

C c— 1 C

c— 1 a2 + raQ2

Residue T—R—C ( r - \ ) ( c - \ ) T — R—C

a2 Residue T—R—C ( r - \ ) ( c - \ ) ( r - l ) ( c - l )

a2

Total T rc— 1 — —

The parameters crR2 and c r c

2 are estimated by * r R T-R-C

c 2 SR — —

and C T-R-C c - l (r - 1) (c - 1)

(5)

(6)

The test of the null hypothesis that <JR2 = 0 is made by computing the

statistic F = ( 7 )

which, if the null hypothesis is true, follows the F distribution with r — 1, (r — 1) (c — 1) degrees of freedom.

A test of the hypothesis that c r c2 = 0 is conducted in a similar manner.

The analysis of variance technique of which the above are examples can be extended to «-fold classifications. The technique is also valuable in ascertaining whether the effect of a factor is maintained over different levels of other factors, i.e., for detecting the presence of interactions. See ANDERSON, R . L . , and BANCROFT, T. A . , Statistical Theory in Research, McGraw-Hill


Book Co., Inc. New York, 1952; FISHER, R. A . , Statistical Methods for Research Workers, 11th edition, Oliver & Boyd, Ltd., London, 1950.

12.1. Design of experiments. The validity of inferences that can be drawn from an experiment depend upon the manner in which the experiment was conducted. In a comparison of a number of objects it has long been realized that identical conditions should be maintained during the testing of all the objects. Often this is not possible, since not all the objects to be compared can be tested in a short enough interval that conditions can be regarded as identical for all tests.

The scheduling of the tests to achieve unbiased estimates of parameters, and maximum efficiency in the testing of the hypotheses that the experimenter set out to test; the search for an optimum combination of a number of factors each of which may be at several levels; or an investigation of the influence of several factors individually and jointly on some physical quantity are treated in COCHRAN, W. G., and Cox, G. M., Experimental Designs, John Wiley & Sons, Inc., New York, 1950; and FISHER, R. A . , The Design of Experiments, 5th edition, Oliver & Boyd, Ltd., London, 1949.

13.1. Introduction. A measurement process is regarded as precise if the dispersion of values is regarded as small, i.e., cr small.

A measurement process is regarded as accurate if the values cluster closely about the correct value c.

The precision of an individual measurement is the same as the precision for the process generating the measurements.

By accuracy of an individual measurement or of an average of n measurements is usually meant the maximum possible error (constant and/or random) that could affect the observed value and is frequently thought of in terms of the number of significant figures to which the value can be regarded as correct.

13.2. Measure of precision. It is suggested that the measure for representing precision of an individual measurement in a set of n measurements be

12. Design of Experiments

13. Precision and Accuracy

(i)


and for the precision of an average,

s (2)

13.3. Measurement of accuracy. Limits on possible constant errors due to known sources of inaccuracy can be set down from a study of the method of conducting the experiment. Such limits on errors represent a judgment on the part of the experimenter and no probability statements can be attached thereto. The setting down of the upper limit and lower limit for a quantity should involve the possible uncertainties due to any constant error plus some component due to random errors.

Suggested form for reporting results of n determinations of some quantity :

Average x = Number of measurements n = s.d. of average s/\/h = Maximum possible constant errors; E, —E' = Lower limit to quantity : x — E' — Ksf^/n = Lower limit to quantity : x + E + Ksj^n =

where K is either Student's ty K = 2 or 3 for approximation to 2a or 3cr limits (for n large), or any other specified constant.

The upper and lower limit to a quantity express the experimenter's best judgment of the range within which the true value of the quantity being measured lies. A value could then be said to be accurate to as many significant figures as the upper and lower limits agree.

14.1. Introduction. If x follows a distribution with mean jii and standard deviation cr, and F(x) is some function of x, then the mean /x F and variance oF

2 of F(x), where x is the average of n values, is approximated by

provided F(x), F\x) are continuous and nonzero in the neighborhood of the point x = fx.

Caution must be exercised in the application of the formula. It is valid when two conditions are satisfied : (a) if the Taylor expansion

14. Law of Propagation of Error

°f = TO! v .2

(1)

(2)

Fx) = FQi) + F'(n) ( * - , t ) + ( * - J u ) 2 + . . . (3) 2!


is such that Fk(fi) for k = 2, 3, . . . is small, and (b) if the expected value of (x — fJi)k, for k = 3, 4, . . . is small (i.e., that the third and higher moments of the distribution of x be small). There are cases where condition (a) is not satisfied, yet the formula is valid because condition (b) holds, or where neither condition holds but the product of Ffc(/x) and the kth moment is small for k > 2.

EXAMPLE : Let x be normally distributed with mean and standard deviation cr. Then the mean and variance of y — x2 are according to the above formula

mean of y = fi2

variance of y = 4fi2a2

The exact answer for the mean is /x2 - f a2, and for the variance, 4/x2a2 + 2o-4. When a is large the " propagation of error formula " can be seriously in error in this case.

EXAMPLE : Let x have mean | = ^ + 1 / 2 o r 2 , and variance i2(e°2 — 1), and with third moment, or average value of (x — £)3 = £\ezo — 2>e° + 2), fourth moment, or average value of (x — £)4 = £ 4 (e 6 a — 4esa + 6ea — 3 ) , etc., such that y = log x is normally distributed with mean \L and standard deviation a. The approximation gives

for the mean of y :

log £ = /x + \a2

for the variance of y :

1 a 2 cr4 a6

If a is large, the approximate formula can be seriously in error, in this case because the third and higher moments of the original distribution are not small.

If H(xlyx2,x3,...) is a function of xly x2, x 3 , . . . which have mean values /xx, /x2, /x3, . . . and variances cr, 2 , cr 2

2 , cr 32 , . . . and correlations p 1 2 , p 1 3 , p 2 3 ,

then the variance, aH2, of H^x^x^x^...) is given by

°H* = [HXi']w + [H-^w + [Hx;]w + • • • ) ( 4 )

— Ipyp^^H^H^ — Ipô-pzH^H^ — 2P23<J2<7sHX2 HH — ... j

all derivatives being evaluated at /xx, /x2, /x3, . . . .

The restrictions on the validity of this formula are analogous to the restrictions applying to F(x).


14.2. Standard deviation of a ratio of normally distributed variables. If z = xjy where x is normally distributed with mean \ix and standard deviation ax; y is normally distributed with mean \LV and standard deviation (jy; the correlation between x and y is p and / x ^ / c r ^ is sufficiently large (say greater than 5) then z will be approximately normally distributed with mean of z .= fijfjby, and standard deviation of

[FIELLER, E. C , Biometrika, Vol. 24 (1932), p. 428.]

14.3. Standard deviation of a product of normally distributed variables. If # and y are normally distributed with means JJLX and py and standard deviations crx and ay, then w = xy will have mean \xx\iy and standard deviation

[CRAIG, C C , Annals of Mathematical Statistics, Vol. 7 ( 1 9 3 6 ) , p. 1.]

(1)

Chapter 3

N O M O G R A M S

B Y D O N A L D H. M E N Z E L Professor of Astrophysics at Harvard University

Director of Harvard College Observatory

1. Nomographic Solutions

1.1. A nomogram (or nomograph) furnishes a graphical procedure for solving certain types of equations, primarily those containing three variables, say a, /?, and y. If any two of these quantities are known the third follows directly from the conditioning equation between the three parameters. The determinant,

yi y2

y*

o

represents the equation : yi-y* _y^-yz

(i)

(2) iAf -J 4/v 2 3

The three points fayi), (x2y2), a n d ( # 3 ^ 3 ) a r e collinear because the two segments possess identical slopes "and because they have a point, (x2,y2) in common.

Now, suppose that we can write

xi=<l>i(0L) yi = llJi(0C) x2 = MP) y2 = h(P) (3)

Equations (3) define three curves, in parametric form, whose points we can label in terms of our basic parameters a, j8, and y. Thus, if we know oc and j8, for example, a straight line from the appropriate points on the oc and curves will intersect the y curve at the value of y that satisfies the equations, as indicated in Fig. 1, which represents the artificial equation

2 In p sin oc + 1.5a cos (77/2)7 + 2P2 + l-5P2 s i n 2 ("I2)?

— 1.5 cos (7r/2)y In p — 4 sin oc — 3 sin oc sin (7r/2)y — ocp2 = 0

141

142 NOMOGRAMS § 1 1

/(.VVN I 3.75 I

3.75 3.£ 1 on 3.£

S > \ 0.75" 3.25 \

S N •s 1 fsn \ \ \ 0.50

\ / 3 \

/ 2,75 M \ / \

2,75 M \

1. / \ V. \ 0 or-1. ( \ \ 1 \ 9 50 \ / 50

9. 9R / s \ V \

2.C \ M _ "2uo 2.C Ns U s s \ / \ \ \ / \ h 5 j \

3 75 J 1 5 )

i_ 2.25 \ / \ 3

75 / a I / \ r. 25 1 \ / S. / I 25

\ / /

K 1 0° 2.50 \ / 3.50 / / \ ,

\ / 0.7b \ 1.75

3.25 \ 5C 3. 00 .5 O

5C 3. 00 .5

\ 25 1 1 U.25

X l.Ou P 0 0

0. / 5

O 5 1 5 /

FIGURE 1

This equation fits the determinant (1), with

xx = 2 sin a, yx = oc

x2=fi2, y2 = lnfi

xs= 1.5 cos (77/2)7, ys = 2.0 + 1 - 5 sin (7R/2)y

The line connecting given points j8 = 1.5 and y = 1 intersects the a curve twice. There are thus two roots, approximately at oc = 1.1 and 2.9. By increasing the number of coordinates marking the values of a, /?, y, we can read as closely as desired. In many examples, of course, some of these curves are straight lines. ,

To represent an equation nomographically we must first find an equivalent determinant which we then transform by standard rules until we obtain our equation in the form of (1), viz.,

§ 1 - 1 NOMOGRAMS 143

<W0 & ( « ) i !

j 4 ( y ) 1 ( 4 )

We must separate the variables, as above. Although this equation satisfies the initial condition, the diagram resulting

from it is by no means the most general one possible. Let a, b, c, d, e, / , g , h, i represent nine arbitrary constants. Then, using the transformation properties of a zero-valued determinant, we obtain the general form, which expansion will prove to be equivalent to (4).

\axf>1 + b]j1 + c tfyj + + f

gfa + hi/*! + i gfa + hi\sx + i

a$2 + bif;2 + c dcf)2 + eifj2 + f gcj)2 + hifj2 + i g</>2 + hi/j2 + i

a</)s + bi/j3 + c # 3 + ^03 + /

1

1

= 0 (5)

\ g(f>3 + hi/js + i gcf)3 + hifj3-

By choosing these constants a — i appropriately, we can control the positions and scales of the a, /?, and y curves in order to achieve maximum accuracy for the desired range of variables.

The diagram representing Eq. (5) is a linear transformation of the diagram representing Eq. (4). In other words, (5) is a mere projection of (4) upon some arbitrary plane. Two of the constants, in effect, fix the scales of x and y . Two more determine the choice of origin, including the possibility of shifting any specific point to infinity. Four more define some specific line in the plane and the remaining constant fixes the degree of rotation of the original diagram about this arbitrary line as an axis. The nine constants thus completely determine the projection scheme. Some experimenting with constants is generally necessary to produce the best diagram for a given problem.

A few examples follow. The simple additive nomogram

/ i ( « ) + / . 0 3 ) + / 8 ( y ) = o

yields the basic determinant:

fl 1 i: fi 1 2 -/. 1 o | = 1 1 = i i

0 I 0 1 -/. 0 1

= 0

(6)

( 7 )

The third determinant defines three parallel straight lines. The y scale fits along the y axis, the f3 scale along the line y = 1, with the oc scale halfway

144 NOMOGRAMS §1.1 between. Now, transforming the determinant by (5) we can get a general form, wherein the three lines may no longer be parallel.

The multiplicative form M°¥tf) =MY) (8)

assumes the form of (6) if we take logarithms. However we often find the following more useful.

IA o l j \A o i+A i h o! = j l h i io -/8 1| |0 1

o 1 1 / , 1

o -/, i = 0 (9)

Here the y scale coincides with the negative part of the y axis. The /? scale extends in the positive direction along y = 1. The oc scale lies along the x axis. By projecting this diagram we can make it assume a Z-shape, with the two arms of the Z not necessarily parallel.

For four variables several alternatives arise. Occasionally the nomograms for four variables are identical in form with those for three, except that one (or more) of the scales are shifted. A sort of sliderule index, to set for the fourth variable, gives the necessary flexibility.

Sometimes we may treat one of the variables as a constant, calculating a different nomogram for each value of the fourth parameter. One can then interpolate between various curves to select the appropriate point.

In still other cases, as for example, in the equations,

M«Vtf) =MYW) =U0 (io)

involving the four variables, we may introduce an arbitrary variable £, as above. Now we can compile two independent nomograms, one involving a, j8, and £ and the other y, S, and f. With the £ scale identical on both, we can make a single diagram with a common £ scale. Then, if we are given, say oc, j8, and y, and wish to find 8, we first join oc and j8 with a straight line that intersects £ at some special point. Then connect this point with the value of S. The intersection of this line with the y axis determines y.

Bibliography

1. D'OCAGNE , M., Traite de Nomographic, 2nd ed. Gauthier-Villars et Cie, Paris, 1921. The standard work and most complete reference.

2. DOUGLASS, R . D . and ADAMS, D . P., Elements of Nomography, McGraw-Hill Book Company, Inc. New York, 1947.

Chapter 4

PHYSICAL CONSTANTS B Y J E S S E W . M . D U M O N D

Professor of Physics, California Institute of Technology

AND E . R I C H A R D C O H E N Physicist, North American Aviation Inc., Downey, California

1. Constants and Conversion Factors of Atomic and Nuclear Physics *

It is an interesting paradox that whereas the physical constants, and particularly those associated with the atom, fixed as they are by nature rather than by man, are believed to be the most invariable things known to science, the best methods of arriving at reliable numerical values for these constants have undergone more change in recent years than has been seen in almost any other branch of physics. Recent rapid advances in new techniques and in our theoretical understanding have resulted in new, entirely different, and very superior approaches for ascertaining the best values. These improvements at each stage usually render quite obsolete most of the methods of the preceding stage or era, not because the earlier methods were wrong in principle, but because they admitted of so much less accuracy in practice that they are no longer worth considering in competition with the newer methods. Recently this pace has been markedly accelerated so that, in the span of only a little over four years, completely new experimental methods growing out of the development of microwave and atomic beam techniques have so greatly improved the precision of our knowledge of the atomic constants as to warrant a complete re-evaluation of the entire situation. There has resulted an improvement in accuracy of approximately tenfold as regards our knowledge of the constants and conversion factors. No guarantee can be given that another two or three years hence an equally radical revolution may not have rendered the present methods equally obsolete. It is a fortunate and reassuring fact, however, that with all these

* Values as of October, 195,5.

145

146 PHYSICAL CONSTANTS

changes in method the results obtained, the values of these constants and conversion factors, give every indication of tending to settle down within narrower and narrower limits of uncertainty toward stable and definite results.

Because of this ephemeral character of the methods of determining these constants, we shall say here only enough about these methods to clarify certain important points bearing on the use of the table of values. The reader who wishes to study our present methods in detail will find them described in papers by the present authors [Rev. Mod. Phys.> 25, 691 (1953); 27, 363 (1955)]. A list of references at the end of this chapter also indicates some of the key papers bearing on this subject over the last twenty-five years. The outstanding work of R. T . Birge during this period in this field has been undoubtedly the greatest single factor in bringing about its present high state of development.

The appended table of values of constants and conversion factors of atomic and nuclear physics was computed by means of formulas (given with each constant in the table), all of which can be expressed directly in terms of the values of the following five key quantities combined with certain other accurately known auxiliary constants. The five key quantities (which we call our " primitive unknowns " ) are N, Avogadro's number; e, the electronic charge; a, Sommerfeld's fine structure constant; c, the velocity of light; and A /Ag, the conversion factor from the Siegbahn arbitrary scale of x-ray wavelengths (x units) to absolute (or " grating " ) wavelengths in milli-angstroms ( 1 0 - 1 1 cm). The computation of the table of values from these five primary values is a straightforward process on which we need not dwell. The computation of the values of the five primitive unknowns from various experimental data is accomplished by the following general method. A number of " observational equations " (in the present instance thirteen) is set up to express the results of experimental measurements. It is a peculiarity of the atomic constants that in but few instances are the directly measured numerical quantities which result from experimental work capable of determining a single primitive unknown (such as e or N). In most cases, it is some function of two or more of the primitive unknowns that experiments determine. The number of such determined independent functions of the unknowns should exceed the number of unknowns as much as possible to afford a considerable degree of overdetermination in order to furnish a check (the only one in fact available) on the interconsistency of the data and its freedom from systematic errors. The mathematical tool especially developed for just such a situation as this is the method of least squares. * This permits us to compute a set of " adjusted " o r compromise values of the

* See Chapter 2 .

PHYSICAL CONSTANTS 147

unknowns that do the least violence to all the input data, according to a systematic rule for minimizing such violence. This rule allows greater latitude for deviation to the less accurate determinations. In the present instance, a judgment of the violence done to the input data by the process of least-squares adjustment can be obtained from the statement that the ratio of our " precision measure by external consistency " (a measure of the compatibility of the overdetermined input equations) to our "precision measure by internal consistency " (based on estimates of the experimental accuracy of the input data) was rejri =

A rather delicate point of novel character arises, in the present least-squares adjustment. In order to minimize the numerical work, which piles up with appalling rapidity as the number of unknowns increases, an effort is made to formulate the observational equations as compactly as possible. A result of this is that, unless care is taken to avoid such a condition, the numerical values (the results of experimental measurements), to each of which is equated some function of the unknowns in each observational equation, may not always be single measurements but may be a function of several independent measurements. If, furthermore, the same independent measurement may have been used as a contributing factor in computing the numerics for several different observational equations, these numerics may thus become " observationally correlated." Great care and attention must be given either (1) to formulate the equations so these correlations are absent or at most insignificant, or (2) to take account of them by an extension of the method of least-squares which has been especially developed for this purpose.

Two scales of atomic weights exist, the physical and chemical scales. On the physical scale the O 1 6 isotope of oxygen has, by definition, the atomic weight 16. On the chemical scale, atomic weight 16 is assigned to the average atomic weight of a mixture of the oxygen isotopes in their naturally occurring abundance ratios. Unfortunately, if we are interested in the highest accuracy, the chemical scale is ambiguous and indefinite now that the precision with which atomic weights can be measured (by mass spectro-graphic means) is amply sufficient to show that the naturally occurring abundance ratios of the oxygen isotopes is different for different natural sources of oxygen (such as water or iron ore on the one hand and limestone or air on the other). All the atomic weights used in the following table are therefore given on the physical scale of atomic weights.

If one wishes to be strictly correct, some care must be used in computing precision measures of quantities depending functionally upon two or more of the numerical values given in the appended tables because these values are


not completely uncorrelated in the observational sense. This unavoidable complication and the simple methods of handling it (by means of correlation coefficients and cross-product contributions to the error measure) are explained in an article, [Rev. Mod. Phys., 25, 691 (1953)]. See.note p. 154a.

2* Tabic of Least-Squares-Adjusted Output Values (November, 1952)

I. AUXILIARY CONSTANTS USED :

These auxiliary constants are quantities which are uncorrelated (observationally) with the variables of the least-squares adjustment.

Rydberg wave number for infinite mass : *

Rm r= 109737.309 ± 0.012 cm" 1

Rydberg wave numbers for the light nuclei:

RH = 109677.576 ± 0.012 cm" 1

i ? D = 109707.419 ± 0.012 cm' 1

RHe* = 109717.345 db 0.012 cm- 1

i ? H e4 = 109722.267 ± 0.012 cm" 1

Atomic mass of neutron : n = 1.008982 ± 0.000003

Atomic mass of hydrogen : H = 1.008142 ± 0.000003

Atomic mass of deuterium :

D = 2.014735 ± 0.000006

Gas constant per mole (physical scale) :

R0 = 8.31696 dz 0.00034 X 107 erg mole"1 deg"1

Standard volume of a perfect gas (physical scale) : V0 = 22420.7 ± 0.6 cm3 atm mole"1

*This differs from the value R^ = 109737.311 dz 0.012 cm" 1 given by E.R.Cohen (Ref. 11) and which was used in the least-squares adjustment. In Ref. 11 a tentative value Nm = 5.48785 X 10~4 for the atomic mass of the electron was used with the proviso, p. 359, that " ... an increase of 1 part per million in the electron mass will produce an increase of 0.00005 c m - 1 in the Rydberg." (The coefficient 0.0005 c m - 1 in the text is a typographical error.) The present value has therefore been revised, using this coefficient to accord with our present output value of Nm and similar modifications have been made in the Rydberg values of the light nuclei.

PHYSICAL CONSTANTS

II. LEAST-SQUARES ADJUSTED-OUTPUT VALUES :

(The quantity following each ± sign is the standard error.)

Velocity of light: c = 299793.0 ± 0.3 km sec"1

Avogadro's constant (physical scale) :

N = (6.02486 ± 0.00016) x 10 2 3 (g mole)"1

Loschmidt's constant (physical scale) :

L0 = NIV0 = (2.68719 ± 0.00010) x 101 9 cm" 3

Electronic charge : e = (4.80286 ± 0.00009) X 10 " 1 0 esu

e' = e\c = (1.60206 ± 0.00003) X 10~ 2 0 emu

Electron rest mass : m = (9.1083 ± 0.0003) x IO" 2 8 g

Proton rest mass : m9 = MJN = (1.67239 ± 0.00004) X I O - 2 4 g

Neutron rest mass : mn = n/N = (1.67470 ± 0.00004) x IO" 2 4 g

Planck's constant: h = (6.62517 ± 0.00023) X 10~2 7 erg sec

ft = h/ln = (1.05443 ± 0.00004) X 10~2 7 erg sec

Conversion factor from Siegbahn #-units to milliangstroms :

Ag/As = 1.002039 ± 0.000014

Faraday constant (physical scale) :

F = Ne = (2.89366 ± 0.00003) X 101 4 esu (gmole) F' = Nelc = (9652.19 ± 0 . 1 1 ) emu (g mole)"1

Charge-to-mass ratio of the electron :

e/m = (5.27305 ± 0.00007) X 101 7 esu gm" e'\m = el(mc) = (1.75890 ± 0.00002) X 107 emu gm"

Ratio hje : hie = (1.37942 ± 0.00002) X 10 " 1 7 erg sec (esu)"1

Fine structure constant :

oc = e2l(hc) = (7.29729 ± 0.00003) x 10~3

l/« = 137.0373 ± 0.0006 a/27r = (1.161398 ± 0.000005) x IO - 3

a 2 = (5.32504 ± 0.00005) X 10~5

1 - (1 - a 2 ) 1 / 2 = (0.266252 ± 0.000002) x 10~4


Atomic mass of the electron (physical scale) :

Nm = (5.48763 ± 0.00006) x 10" 4

Ratio of mass of hydrogen to mass of proton : *

[ Nm I - 1

1 - — (1 - W ) \ = 1.000544613 ± 0.000000006 Atomic mass of proton (physical scale) :

MV = U - Nm = 1.007593 ± 0.000003

Ratio of proton mass to electron mass : MJ(Nm) = 1836.12 ± 0.02

Reduced mass of electron in hydrogen atom :

(JL = w M p / H = (9.1034 ± 0.0003) x 10~2 8 g

Schrodinger constant for a fixed nucleus : Imjh2 = (1.63836 ± 0.00007) X 102 7 erg"1 cm" 2

Schrodinger constant for the hydrogen atom :

2^/ft2 = (1.63748 ± 0.00007) X 102 7 erg"1 cm" 2

First Bohr radius :

a0 = h2l(me2) = aK^R^) = (5.29172 ± 0.00002) X 10 ' 9 cm

Radius of electron orbit in normal H 1 , referred to center of mass :

a0' = «0(1 - « 2 ) 1 / 2 = (5.29158 ± 0.00002) x 10~9 cm

Separation of proton and electron in normal H 1 :

V = aQ'RoolRu = (5.29446 ± 0.00002) X 10" 9 cm

Compton wavelength of the electron : Xce = h\mc) = a 2 / (2 i?oo ) = (24.2626 ± 0.0002) X 10 " 1 1 cm

\e = Kel(2ir):= (3.86151 ± 0.00004) X 1 0 - l l c m

Compton wavelength of the proton :

AC2> = hjmPc) = (13.2141 ± 0.0002) X 10- 1 4 cm K = KJ2TT) = (2.10308 ± 0.00003) x 10" 1 4 cm

Compton wavelength of the neutron :

Kn = hl(mnc) = (13.1959 dz 0.0002) x 10~1 4 cm . = Knl(27r) = (2.10019 ± 0.00003) X 10~1 4 cm

Classical electron radius :

r0 = e2l(mc2) = a 3/(477JR0 0) = (2.81785 dz 0.00004) X 10~1 3 cm r 2 = (7.94030 ± 0.00021) X 10 " 2 6 cm

* The binding energy of the electron in the hydrogen atom has been included in the quantity. The mass of the electron when found in the hydrogen atom is not w, but more correctly m(l — %oc2 + ...).


Thompson cross section :

( i ) rrr02 = (6.65205 ± 0.00018) X IO" 2 5 cm2

Fine-structure doublet separation in hydrogen :

/ 1 \ r <*' / 5 5.946 \ i

= 0.365871 ± 0.000003 cm" 1

= 10968.56 ± 0.10 Mc sec"1

Fine-structure separation in deuterium :

AED = AJE"H RDIRU = 0.365970 ± 0.000003 cm" 1

= 10971.54 i 0.10 Mc sec-1

Zeeman displacement per gauss :

(elmc)l(47Tc) = (4.66885 ± 0.00006) X 10~5 cm" 1 gauss"1

Boltzmann's constant:

k = RJN = (1.38044 ± 0.00007) x IO - 1 6 erg deg"1

k = (8.6167 ± 0.0004) X 10~5 ev deg-1

\jk = 11605.4 ± 0 . 5 deg ev" 1

First radiation constant :

cx = %-nhc = (4.9918 ± 0.0002) x I O 1 5 erg cm

Second radiation constant:

c2 = hc/k = 1.43880 ± 0.00007 cm deg

Atomic specific heat constant :

c%\c = (4.79931 ± 0.00023) X 10" 1 1 sec deg

Wien displacement law constant : *

AmaxT = c2/(4.96511423) = 0.289782 ± 0.000013 cm deg

Stefan-Boltzmann constant :

a = (TT 2 /60) (tfjWc2) = (0.56687 ± 0.00010) X 10~4 ergs cm" 2 deg~4 sec"1

Sackur-Tetrode constant :

(S0IR0)Ph = *- + ln(27r J R 0 ) 3 / 2 / ? -W- 4 = - 5.57324 ± 0.00007

(So)vn = ~ (46.3524 ± 0.0020) X 107 erg mole"1 deg"1

Bohr magneton :

^ = heKAimic) = ieXce = (0.92731 ± 0.00002) X I O - 2 0 erg gauss"1

* The numerical constant 4.96511423 is the root of the transcendental equation, x = 5(1 - e~x).


Anomalous electron-moment correction :

[ i + _ ! L - 2 . 9 7 3 -vl = nJto = 1.001145358 dz 0.000000005 L (2*-) 7T2 J

(Computed using adjusted value a = (7.29729 ± 0.00003) X 10 " 3 .)

Magnetic moment of the electron :

fie = (0.92837 ± 0.00002) X 1 0 " 2 0 erg gauss"1

Nuclear magneton: ^ = hej(Airmvc) = fi0NmlH+ = (0.505038 ± 0.000018) X 1 0 " 2 3 erg gauss"1

Proton moment:

\i = 2.79275 dz 0.00003 nuclear magnetons

= (1.41044 dz 0.00004) X 1 0 " 2 3 ergs gauss"1

Gyromagnetic ratio of the proton in hydrogen, uncorrected for diamagnetism :

/ = (2.67523 ± 0.00004) X 104 radians sec"1 gauss"1

Gyromagnetic ratio of the proton (corrected) :

y = (2.67530 ± 0.00004) X 104 radians sec"1 gauss"1

Multiplier of (Curie constant)1/2 to give magnetic moment per molecule :

(3&/iV)1/2 = (2.62178 db 0.00010) X 1 0 " 2 0 (erg mole deg" 1 ) 1 / 2

Mass-energy conversion factors :

1 g = (5.61000 db 0.00011) X 10 2 6 Mev

1 electron mass = 0.510976 ± 0.000007 Mev

1 atomic mass unit = 931.141 dz 0.010 Mev

1 proton mass = 938.211 dz 0.010 Mev

1 neutron mass = 939.505 ± 0.010 Mev

1 ev = (1.60206 dz 0.00003) X 1 0 " 1 2 erg Ejv = he = (1.98618 dz 0.00007) X 10" 1 6 ergcm E\g = (12397.67 dz 0.22) X 10" 8 ev cm E\s - 12372.44 ± 0.16 kilovolt ^c-units E\v = (6.62517 ± 0.00023) X 10 " 2 7 erg sec Ejv = (4.13541 ± 0.00007) X 10" 1 5 evsec v\E == (5.03479 ± 0.00017) X 10 1 5 cm" 1 erg"1

v\E = 8066.03 ± 0.14 cm" 1 ev" 1

vjE = (1.50940 ± 0.00005) X 102 6 sec"1 erg"1

v\E (2.41814 ± 0.00004) X 10 1 4 sec"1 ev" 1


Neutrons:

de Broglie wavelengths, AD, of elementary particles : *

Electrons :

AD e = (7.27377 ± 0.00006) cm2 secrVy

= (1.552257 ± 0.000016) x I O - 1 3 cm (erg)1/21(E)1'*

= (1.226378 ± 0.000010) X IO"7 cm (ev) 1 / 2 / (£) 1 / 2

Protons:

AD j ) = (3.96149 ± 0.00005) X IO"3 cm2 sec^/v

= (3.62253 ± 0.00008) X 10 " 1 8 cm (erg)1 / 2 /(E)1 / 2

= (2.86202 ± 0.00004) X IO"9 cm (ev) 1 / 2 / (E) 1 / 2

AD n = (3.95603 ± 0.00005) X IO"3 cm2 sec^/v

= (3.62004 ± 0.00008) X IO" 1 5 cm (erg)1?21(E)112

= (2.86005 ± 0.00004) X IO"9 cm (ev)1''21(E)1''2

Energy of 2200 m/sec neutron :

E 2 2 0 0 = 0.0252973 ± 0.0000003 ev

Velocity of 1/40 ev neutron:

v0.025 = 2187.036 ± 0.012 m/sec

The Rydberg and related derived constants :

Roo = 109737.309%: 0.012 c m - 1

Root = (3.289848 ± 0.000003) X 101 5 sec"1

Ra,hc = (2.17958 ± 0.00007) X lOêrgs

Roo^"1 x IO"8 = 13.60488 ± 0.00022 ev

Hydrogen ionization potential:

/ he2 \ I a2 \ /0 = RH - j ( J + J + - ) x 1 0 - 8 = 1 3 - 5 9 7 6 5 ± 0.00022 ev

* These formulas apply only to nonrelativistic velocities. If the velocity of the particle is not negligible compared to the velocity of light, c, or the energy not negligible compared to the rest mass energy, we must use AD = Ac[e(e ± 2 ) ] - 1 / 2 , where Ac is the appropriate Compton wavelength for the particle in question and e is the kinetic energy measured in units of the particle rest-mass.


Bibliography

ByR. T. BIRGE Revs. Modern.Phys., 1, 1 ( 1 9 2 9 ) . * Phys. Rev., 40, 2 0 7 ( 1 9 3 2 ) . Phys. Rev., 40, 2 2 8 ( 1 9 3 2 ) . Phys. Rev., 42, 7 3 6 ( 1 9 3 2 ) . Nature, 133, 6 4 8 ( 1 9 3 4 ) . Nature, 134, 7 7 1 ( 1 9 3 4 ) . Phys. Rev., 48, 9 1 8 ( 1 9 3 5 ) . Nature, 137, 1 8 7 ( 1 9 3 6 ) . Phys. Rev., 54, 9 7 2 ( 1 9 3 8 ) . Am. Phys. Teacher,!, 3 5 1 ( 1 9 3 9 ) . Phys. Rev,, 55, 1 1 1 9 ( 1 9 3 9 ) . Phys. Rev., 57, 2 5 0 ( 1 9 4 0 ) . Phys. Rev., 58, 6 5 8 ( 1 9 4 0 ) . Phys. Rev., 60, 7 6 6 ( 1 9 4 1 ) . * Reports on Progress in Physics, London, VIII, 9 0 ( 1 9 4 2 ) . * Am. J. Phys., 13, 6 3 ( 1 9 4 5 ) . * Phys. Rev., 79, 1 9 3 ( 1 9 5 0 ) . Phys. Rev., 79, 1 0 0 5 ( 1 9 5 0 ) .

OTHER REFERENCES 1. F . G . DUNNINGTON, Revs. Modern Phys., 11, 6 5 ( 1 9 3 9 ) . 2 . J . W . M . D U M O N D , Phys. Rev., 56, 1 5 3 ( 1 9 3 9 ) . 3. Phys. Rev., 58, 4 5 7 ( 1 9 4 0 ) . 4 . F . KIRCHNER, Ergeb. exakt. Naturwiss., 18, 2 6 ( 1 9 3 9 ) . 5. J. W. M . D U M O N D and E. R. COHEN, Revs. Modern Phys., 20, 8 2 - 1 0 8 ( 1 9 4 8 ) . 6 . J . W . M . D U M O N D , Phys. Rev., 77, 4 1 1 ( 1 9 5 0 ) . 7. J. W. M . D U M O N D and E. R. COHEN, American Scientist, 40, 4 4 7 ( 1 9 5 2 ) . 8. J. A. BEARDEN and H. M . W A T T S , Phys. Rev., 81, 7 3 ( 1 9 5 1 ) . 9. BEARDEN and W A T T S , Phys. Rev., 81, 1 6 0 ( 1 9 5 1 ) .

10 . E. R. COHEN, Phys. Rev., 81, 1 6 2 ( 1 9 5 1 ) . 1 1 . E. R. COHEN, Phys. Rev., 88, 3 5 3 ( 1 9 5 2 ) . 1 2 . J. W . M . D U M O N D and E. R. COHEN, Rev. Modern Phys., 25, 6 9 1 ( 1 9 5 3 ) . 13 . E. R. COHEN, Revs. Modern Phys., 25, 6 9 1 ( 1 9 5 3 ) . 14 . E. S . DAYHOFF, S . TRIEBWASSER, and W . LAMB JR., Phys. Rev., 89, 1 0 6 ( 1 9 5 3 ) . 15 . S . TRIEBWASSER, E. S . DAYHOFF , and W . LAMB JR., Phys. Rev., 89, 9 8 ( 1 9 5 3 ) . 16 . E. R, COHEN, J , W . M , D U M O N D , T. W , LAYTON , and J . S . RQLLETT, Rev. Mod.

Phys., 27, 3 6 3 ( 1 9 5 5 ) , 17 . E. R. COHEN and J , W , M , D U M O N D , Phys, Rev. Lett., 1 , 2 9 1 ( 1 9 5 8 ) . 18 . A. PETERMANN, Helv. Phys. Acta., 30, 4 0 7 ( 1 9 5 7 ) . 1 9 . C . M . SOMMERFIELD, Phys. Rev., 107, 3 2 8 ( 1 9 5 7 ) . 2 0 . J . W . M . D U M O N D , Boulder Conference on Electronic Standards and Measurements*

(Paper 1 2 ) (August 13 , 1 9 5 8 ) ,

* Longer and more important review articles. The first is a classic of considerable importance since it reviews carefully all the constants of physics rather than just the atomic constants. The numerical values of the atomic constants given therein, as well as the methods of determining them, have, however, undergone considerable modification since it was written.

PHYSICAL CONSTANTS 154a.

NOTE : In 1 9 5 7 a recalculation of the anomalous magnetic moment of the electron by Petermann (Ref. 1 8 ) and Sommerfield (Ref. 1 9 ) showed that the previously accepted value was in error by approximately 1 4 ppm. The effect of this change has beer* discussed in the literature (Ref. 1 7 , 2 0 ) , but since a complete re-evaluation has not been carried out, it is felt that data as of 1 9 5 5 given in the tables above should stand until such time as the new evaluation is available. The major corrections are indicated in Ref. 1 7 .

Chapter 5

CLASSICAL MECHANICS B Y H E N R Y Z A T Z K I S

Assistant Professor of Mathematics

Newark College of Engineering

The formulas of classical mechanics lie at the base of practically all major fields of modern physics. Although the methods of quantum mechanics are necessary for the understanding of microscopic systems, the methods of even that field are best interpreted in terms of the formulas of classical mechanics, by way of contrast. The following set of formulas, although far from complete, represents an attempt to select from the enormous number of formulas of classical mechanics those that have a particularly modern application.

Notation : A vector quantity will be indicated by bold-face italic type, e.g., a. The symbol a will represent the magnitude of a. The Cartesian components of a will be denoted by either aly a2, a3, or ax, ayy az. The scalar product between two vectors will be denoted by a • b> and the vector product by a x b. Cartesian coordinates will be denoted by x,'y, z, or xly x2y x3, and time by t. The total time derivative will occasionally also be denoted by a dot over the quantity. The total time derivative of a vector quantity wxyyyzyt) is given by

dw dw / ,x

where v is the velocity with which the material point, which is the carrier of the property wy moves.

1. Mechanics of a Single Mass Point and a System of Mass Points

l . L Newton's laws of motion and fundamental motions. a. Bodies not subject to internal forces continue in a state of rest or a

straight line uniform motion (law of inertia). The equations of motion are

d H = d 2 y = d ' 2 z = 0 m dt2 dt2 dt2 U K

155

156 CLASSICAL MECHANICS

Any coordinate system in which Eq. (1) holds simultaneously for all masses, not subject to forces, is callecj an inertial coordinate system. The collection of all reference frames which move with constant velocity with respect to a given inertial system forms the totality of all inertial frames. The coordinate system anchored in the center of mass of the fixed stars is a good approximation of an inertial system.

b. Bodies which are subject to forces undergo acceleration. The acceleration is parallel to the force and its magnitude is the ratio of the force acting on the body and the (inertial) mass of the body. The mathematical form of the second law is

f=ma ( 2 )

or A = mx\ f2 = myt f3 = mz (3)

(force equals mass times acceleration.)

c. T o each force exerted by a mass point A on another mass point B corresponds a force exerted by B on A. The second force is equal in magnitude to the first and opposite in direction (law of action and reaction).

p = mv [v=(x,yiz)]

is called the linear momentum. The force law can also be written

The acceleration a can also be decomposed into a component tangential to the path

and a component normal to the path

h = R\v* ( 6 )

where R is a vector that has the direction and magnitude of the radius of curvature. The expression mbn is called the centripetal force.

If r is the position vector of the mass point w, the vector product m(r x v;) is called the angular momentum, and if the force / a c t s on the mass, the vector product ( r x / ) is called the torque (or moment of force) inerted by the force / .

The work dW done by the force / during a displacement dr is defined by

(7)

§1 .2 CLASSICAL MECHANICS 157

1 p2

T = — mv2 = ~~ is called the kinetic energy.

dW —j- = f • v is called the power of the force / .

1.2. Special cases, a. Central force. If the force f(r>t). is parallel to r, then / x r = 0, and therefore

m^(r x r) = m(r x r) = 0 (1)

(law of conservation of angular momentum). The vector | A: = — - — is a constant, nonlocalized vector (area! velocity vector). In planetary motion Eq. (1) expresses Kepler's second law.

b. Mass point in a potential field. If the force / can be written as the negative gradient of a scalar function U(r,t), i.e., if

/ = - g r a d C / M (2)

U is the potential energy. If U does not depend implicitly on time, i.e.,

U=U(r) (3)

the force / is called conservative, and

(T + U) = 0 (conservation law of energy) (4)

or T + U = E = constant = total energy.

c. Constraints. The mass points may be constrained to move along certain surfaces or curves, called constraints. These constraints can be replaced by constraint forces / * which keep the bodies from leaving these constraints but do themselves no work, since the bodies always move at right angles to the constraints. The law of motion for a single mass point under the combined influence of an internal force/and a constraint force / * is

/ + / * = ma (5) If the constraint is given by

#* ,y ,*) = 0 (6)

(i.e., the particle is constrained to move on a surface), the force of constraint is

/ * = Agrad<£ (7)

The Lagrangian multiplier A is an unknown function of space and time.

158 CLASSICAL MECHANICS §1.2

For example, consider a bead that slides frictionless on a straight wire that makes an angle with the horizontal whose tangent is A. We assume the wire in the x — z plane with the positive z axis directed upward.

The equation of the constraint is

z-Ax-B = 0 (*>

The equations of motion are, therefore

mx = — AX (9)

mz = — mg (10)

We can eliminate A from both equations and obtain

— -^x = mz + mg (11)

But since z = Ax (12)

m we have — — x = mAx + mg (13)

t2

or x = — sin oc cos ocg — (14)

t2

z = — sin2 ag y (15)

m and A = j x =mg cos 2 a (16) where oc = arc tan A, and where we assumed the particle started from rest at the origin.

d. Apparent or inertial forces. All the above laws refer to inertial systems. If we refer the equations of motion to accelerated systems, additional terms appear which are called apparent or inertial forces.

The most important cases are :

1. Linearly accelerated systems :

r =#•* + *(*) (17)

§ 1 . 3 CLASSICAL MECHANICS 159

(r refers to the position vector in the inertial frame and r* to the accelerated frame). The equation of motion is

my* = f— mb (18)

where / is the external force and —mb is the inertial force.

2. Rotating coordinate systems : Let A: be a unit vector in the direction of the fixed axis of rotation and co the angular velocfty. The equation of motion is

mr*=f-2ma>(kxr*)-mw2[kx(kxr*)] (19)

— 2mto(k X r*) is the Coriolis force.

— moj2[k x(kx i**)] is the centrifugal force.

1.3. Conservat ion laws. We shall assume a system of N mass points which exert central forces on one another and which are in addition subject to external forces f^(n = 1,2,...,N), where fn* is the force acting on the nth mass point. The following notations will be used.

velocity of the nth mass point

linear momentum of the nth mass point

total linear momentum

total angular momentum

total mass of system

position of center of mass

total external force

All quantities are assumed to be taken with respect to a Cartesian inertial reference frame.

mnVn = Pn

N

n=l

n=l

N M=^mn

N

N

CLASSICAL MECHANICS

The following three conservation laws hold

f = M * = , * (1)

| = % x P ) (2)

J § = f f n * - V n (3)

If the internal forces vanish, the linear momentum, angular momentum, and energy of the system remain constant, hence the name " conservation laws."

1.4. Lagrange equations of the second kind for arbitrary curvilinear coordinates. The equations of motions containing the forces of constraint (see § 1.2c) contain supernumerary coordinates, since the constraints actually reduce the number of degrees of freedom. They also use Cartesian coordinates. Often these Lagrange equations of the first kind (as they are called) are difficult to use. The generalized coordinates qlt qfy w h e r e / is the true number of degrees of freedom, are often more convenient. All the qk must be independent of each other, and

xi = xi(ql9...iqf) j

y i = y49i,-',<it) («•= 1,2, . . . , # ) ( l )

s=l C(is

etc. The total kinetic energy

1 N

L i=l

expressed in terms of the generalized velocities qy will be a homogeneous quadratic function of the q, where the coefficients will in general be functions of the q. We assume the system conservative and denote the potential energy by U and define the Lagrange function or the Lagrangian L to be

L^T-U (3)

§ 1.5 CLASSICAL MECHANICS 161

The Lagrange equations of the second kind read

d_ dt

The quantities pk = dh\dqk are called the generalized momenta.

Note: Equation (4) assumes that all forces are conservative. If there exist in addition nonconservative forces Fk (e.g., magnetic forces or friction forces), the Lagrange equations are to be generalized to

Sometimes we can find a function M of q and q so that even the non-conservative forces Fk appear in the form

dtdqk dqk-b* ( 6 )

In that case we introduce a new Lagrangian L• = L — M , and the equation of motion again assumes the form ( 4 ) :

d dL 3L „ dt dqk 3qk

Consider the motion of an electric charge e in an electromagnetic field. In that case

M = j-(A-v) ( 8 )

where A is the so-called magnetic vector potential and v is the velocity of the charge e.

L5* The canonical equations of motion. If in the Lagrangian L the generalized velocities q are expressed in terms of the generalized momenta/), we obtain a function L* which is numerically equal to L :

L*(4k>PkJ) = Lfak>qk,t) (1)

We introduce a new function H(qk,pk,t) defined by

H(qk,pk9t) == —L*(qk,pk9t) + ^P4lqk>pkj) (2)

The equations of motion (§ 1.4-5) become then 8H


dH (4)

where H the Hamiltonian,

H= T+ U (5)

hence equal to the total energy in a conservative system. The symbolical 2/-dimensional space of q and p comprises the " phase space.'' Through each point in phase space passes exactly one mechanical trajectory. The /-dimensional subspace the coordinates q is called the configuration space.

1.6. Poisson brackets . If F is any dynamical variable of the mechanical system, e.g., the angular momentum, assumed to be expressed in terms of </, py and t, its time rate of change is given by

dF_ ^ /_3F . dF. \ dF__ ^ / dF 3H 8F dH\ 3F dt " 2j \dqk

q k + dpkPk) + dt ~ % \3qk dPk dpkdqk) + dt

is called a Poisson bracket. The Poisson bracket of any two dynamical variables F and G is defined by

An expression such as

Equation (1) can be written

(3)

In particular, (4)

or dH dt

dH dt (5)

In a conservative system, in which the energy does not depend explicitly on time, H is constant. Any constant of the motion (or " integral of the motion " ) has a vanishing Poisson bracket with H.

and


H(qk)pkit) dt = 0 [qk = qk(qk>Pk,t)] (3)

The time and the endpoints are again held fixed.

The Poisson brackets satisfy the identities

[F,G] + [G,F]^0 (6)

F,[G,M]J + [G,[M,F]J + [ M , [ F , G ] J - 0 (7)

The latter is called Jacobi's identity. The Poisson brackets of the canonical coordinates q and p themselves have the values

[9k,qi] = 0 ) [P*>Pi] = 0 (8) [<lk>Pk] = O k i )

These Poisson brackets are of fundamental importance in the Heisenberg formulation of quantum mechanics, where they express the fundamental uncertainty principle.

1.7. Variational principles, a. Hamilton's principle. The Lagrange equations (§ 1.4-4) are the Euler-Lagrange equations of the variational principle

oSô(hLdt = 0 (1) J h

The variation is taken in configuration space only, subject to the restriction that the time (and therefore also the end points of the path) is not varied. This formulation of the Lagrange equations shows their invariance in any coordinate system i.e., if L(q,q,t) == £*(<?*,<?*,/), then in the new coordinate system the same equations hold again ;

S= ^LiqA^dt

is called the action function, or Hamilton's principal function, and (1) defines Hamilton's " principle of least action," the analogue of Fermat's principle in optics, (see § 1.11).

b. Variational principle in phase space. The canonical equations of motion (§ 1.5-3.4) are the Euler-Lagrange equations of the variational principle in phase space (i.e. q and p are considered the coordinates of a point) :


\ sv_ r ev „ „ 8V (4)

V=V(q,p,t) j

8V - 8V „ 0 8V t (5)

V=V(q,p,t) j

8V 8V „ 0 8V ( (6) p«=-w*'' q « = - w k ' ' H = H - ^ )

V=V(p,p,t) j

8V . 8V „ -- 8V i (7)

1.8. Canonical transformations. The form of the equations is independent of any specific coordinate system. The basic principle serves as a unifying guide for expressing more general theories, e.g., in the general theory of relativity or quantum mechanics.

Any transformation law from the canonical COORDINATES q and p to new canonical coordinates q and p, by which the canonical equations remain invariant, is called a canonical transformation. Mathematically the canonical transformations are a special case of the contact transformations. The necessary and sufficient condition that the transformation be canonical is that the two linear differential forms

A / Z/PiAk — H(qkipkit)dt and 2/pkdqk — H(qk,pk,t)dt k=l k=l

differ by a total differential, denoted by dV :

(Hpkdqk-Hdt)-(Lpkdqk-Hdt) = dV (1)

<llc= £*(M2»-ft/>2-»*) ( 2 )

Pk=pjc(qiq2-'Pip2--t) P )

where V is called the generating function. The canonical transformations are particular mappings of phase space upon itself. The following choices are possible for the function V.

V=V(q,q,t)


and />„» + i > 22 = A 2 + ^ - + f2 ^ r 2 sin2 0

Rotation of a Cartesian coordinate system (xvx2,x3) to (xi,x2,xa):

V=I,a(,cxipk ( i , * = 1 ,2 ,3 ) ( 1 1 )

where S a ^ a ^ = 8/ A.

3,

R=l R=l A frequently used transformation is

T7 m 2 4 . - -

s m .q't • P = V2mcop cos g

( 1 2 )

and p2\2m + moj2q2/2 goes over into co/>. This last transformation shows how we can easily find the equations of motion of the linear harmonic oscil-

For example,

V = qiPi + 92p2+ (8)

leads to the identity transformation

4i = <7i Pi=Pi etc.

V = pxr cos (f> + pyr sin ^ + pzz (9)

gives us the transformation from Cartesian to cylindrical coordinates :

x = r cos (f>, pr = px cos (j> + py sin <f>

y — r sin^, pz = —pxr sine/)-\-pyr coscf)

Z = Zy Pz=Pz

The expression ^ + py2 goes over into ^ r

2 + p2\r2.

If we use V = pxr cos 0 sin 6 + / ^ r sin cf> sin 6 + pzr cos # (10)

we obtain the transition to polar coordinates :

x = r cos c/> sin 0, pr = px cos </> sin 6 + ^ sin < sin # + ^ z cos 9

y = r sin <f> sin #, />z — —pxr sin </> sin 0 + / V cos </> sin 6

z = r cos 6y pe = pxr cos <j> cos &• + sin <f> cos 0 — />2r sin 6


lator. Since T = mq2/2 and U = kq2/2 = mco2q2j2, we obtain for the Hamiltonian the expression

2m ^ 2 H

We shall call the new canonical variables <f> and oc and thus have the relations :

Q—\—— sin<£ p = \/2mujoi cos d>

Hence the new Hamiltonian is H = coot, and the equations of motion are

oc — constant <j> = cot + j8

sin (co£ + j8)

The canonical transformations were characterized by the fact that they leave a certain integral in phase space, the action integral, invariant. Still other quantities that are invariant in phase space under canonical transformations, include the so-called integral invariants. They are

Fi=\\i,dPlcdqk (13)

taken over an arbitrary two-dimensional submanifold in phase space. Similarly

Fa= HHXdp&rtidqt (14)

where the summation is taken over every combination of two indices and the integral is again taken over any arbitrary four-dimensional submanifold in phase space. Similarly

^ 3 = HI / / / 2 dp^,ApMAiMx (15)

where the summation is taken over every combination of three indices and the integral is taken over any arbitrary six-dimensional submanifold. The last integral in this sequence is

Ff = ' J . . . fdpxdp2 ... dpfdqxdq2 ... dqf (16)

The last mentioned integral invariant, known as Liouville's theorem in statistical mechanics, says that the volume in phase space is an invariant. The integral invariants were of great significance in the earlier Bohr-Sommerfeld formulation of quantum theory. They were those dynamical quantities of classical mechanics which had to be quantized.

§1.9 CLASSICAL MECHANICS 167

1.9. Infinitesimal contact transformations. * A transformation

h = pk + *Mqi<iz-PiPz-) .» ^~l'z'-'J) (!)

is called infinitesimal if € is a small quantity such that its higher powers can be neglected as compared to the first power. Thus if we have a dynamical function F(qkypk)y we can write

F(qjc + &f>k,pk + €0*) - F(qkfpk) + € X T T - ^ r + * £ fr (2) r=l Vr r = = l °Pr

This infinitesimal transformation will be canonical, if

HprSqr — HprSqr = SF

or 2/>rS<?r - Zpr + ( S 9 r + e§^r) = SF

or omitting terms containing e2, if

— eZprh<j)r + iprfyr = 8^

If we put F = eWy we obtain

or

If we put

then

i r + ^ = y

a

a?r

apr

ap r

= — —-, < = ——

(3)

(4)

(5)

(6)

* See, e. g., HAMEL, G . , Theoretische Mechanic Julius Springer, Berlin, 1 9 4 9 , p. 2 9 9 .


Therefore the infinitesimal canonical transformation is

dV

qk - qk Âqk = €— (7)

3V p k - p k = kpk = - € — (8)

C0Lk where V is the generating function. Therefore we can regard the canonical equations

j 8H j j j 8H , aqk = 7 ~ - dt and dpk = — — dt

as infinitesimal transformations with the Hamiltonian H as the generating function. Any finite canonical transformation can be built up out of successive infinitesimal transformations.

Any arbitrary dynamical function F = F(q,p) undergoes during an infinitesimal canonical transformation, a change AF.

( 9 )

\dqr dpr dpr dqj

This equation brings out again the meaning of the Poisson bracket. If V is the Hamiltonian and if F is a constant of the motion,

A F = 0 (10)

L10* Cyc l i c variables . The Hamilton-Jacohi partial differential equation. If the Hamiltonian H does not contain a certain coordinate, e.g., qlt

H=H(plyq29p2,...,qf,pf,t) (1)

then Pi = 0 (2)

px = constant (3)

Therefore px is a constant of the motion. Helmholtz has introduced the name of cyclic coordinate for such a " hidden " coordinate. We shall try to find such a canonical transformation wherein all the new coordinates are cyclic. We denote the original canonical variables by qk and pk and the corresponding new cyclic canonical coordinates and momenta, respectively, by <f)k and ock. Denote the generating function by S = S(ql9...ioL1,...,t).

§ 1 . 1 0 CLASSICAL MECHANICS 169

Then 8S A , 3S

(compare § 1.8, Case c). By appropriately disposing of the time dependence of the generating

function, we can always make the new Hamiltonian H vanish.* Therefore

H.H + f - 0 ( 5 )

This is the Hamilton-Jacobi partial differential equation. Since the <f>k are cyclic variables, the equations of motion can be written down at once :

otk = constant (7)

and tf>k = tokt + fik (8)

8 S where cok = -— (9)

dock

Since pk = dS/dqk, we have

dS = Jl-^dqk + -§dt=Xp«<i*dt + ^ d t = 2Tdt + ^ d t k=l j.k

We assume furthermore a conservative system. Therefore H = E or 3S/dt = ~E; since E = T + Uy

dS = 2Tdt-Edt = (T-U)dt = Ldt \

rt (10) and S=jt

2Ldt J

This shows that for a conservative system the generating function S is identical with the action function introduced in § 1.7, S — S(qv...,qfft) which we can write

S = S*(qly...yqf,ot1,...yOLf)-Et (11)

3S' as*

Here S* is sometimes called the " reduced " action function because it

* See, e.g., BERGMANN , P. G., Basic Theories of Physics: Mechanics and Electro-dynamics, Prentice-Hall, Inc., New York, 1949, p. 38.


depends only on the coordinates but not on the time. The Hamilton-Jacobi partial differential equation can also be written

as* ds*

^ i ^ - - - , - ^ , - ^ - , - . . ) = W(ocly*2y...yocf) = E (13)

dW

co, = - (14)

An important special case arises if the Hamiltonian is " separable," i.e., if

H=H1(q1,p1) + H2(q2,p2) + ... + Hf(q„pf) (15)

Set Hk(qk,pk) = Wk

where Wx + W2 + ... + Wf = W (16)

If, furthermore,

S*.(q»...4t) = + .... + Sf*(qf) (17)

dS* dSk*

t h e n P ^ ^ = t (18)

and instead of the Hamilton-Jacobi partial differential equation we have / ordinary differential equation to solve, each of the type

(^f) 2+i^*) = r (19) The total energy E of the system is then a function of the oel9 ocf :

E = E(ccv...y*f) (20) As an example, consider the motion of a mass point in a central force field U(r). Using the canonical transformation for spherical coordinates, we obtain

We put S* = Sr*(r) + Se*(d) + S *(<(,) (22)

and the Hamilton-Jacobi equation


dS4 * : a , (24) d<f>

(- H + " + 2 » W - ^ ] = 0 (26)

^f = Pr=^2m[W-U(r)]-^ (27)

- ^ - = P* = <H (29)

The three constants of integration are ocz = pz = mr2 sin2 the angular momentum component along the polar axis; oce, the total angular momentum (o^ = ocd cos y) where y is the angle between the orbital plane and the equatorial plane); and W = E, the total energy.

1.11. Transition to wave mechanics. The optical-mechanical analogy. The Hamilton-Jacobi partial differential equation is of central importance in the transition from classical to wave mechanics. Before discussing the analogy with wave optics we shall discuss the relation to the limiting case of wave optics, viz., geometrical optics. Let /x denote the index of refraction, which may be a function of the coordinates. The paths of the light rays are governed by Fermat's principle :

S (p2 fJL ds = 0 or 8 P*2 — - = 0 (u = phase velocity) (1) J Pi J Pi u

We shall now consider all trajectories of a particle belonging to the same constant total energy E. According to Hamilton's principle

0 = 8 j * 2 IT- E)dt = 8 J * 2 2 T d t = 8 J ' 2 mv2dt = 8 jp2mvds

or 8 | 2 v ds = 0 J/>i

(2)

falls apart into three ordinary^differential equations :

172 CLASSICAL MECHANICS §1 .11

This equation is of the same structure as (1), but the material velocity (or " group velocity " ) v is inversely proportional to the phase velocity u.

liE-U) (3)

The index of refraction is defined by

uo v E — U

(" Vacuum " is here defined as the region where the potential energy U = 0). Thus we can associate with a moving mass point a " wave "whose phase velocity is inversely proportional to the velocity of the particle and whose index of refraction is'y/(E — U)JU. It remains now to recover the " wave equation." We shall use Cartesian coordinates. We assume again a single mass point with the total energy E moving in the potential field U. The action function S = S(x,y,z,t) represents at every instant t a surface in space on which the value of S is constant. Let the velocity with which this surface of a fixed value for S propagates itself in space be denoted by u. An observer travelling with the same velocity u will not see any change in value of S. Therefore

(5) dS

or — — - = u • p = mu • v

But " § - * (6)

and E = mv u (7) But u is parallel to v since u is perpendicular to the surfaces S = constant and p == grad S, which means that p is also perpendicular to the surfaces S = constant. Therefore E — mvu (8)

or W = A _JL== (9) mv V2m(E - U)

If we solve Eq. (9) for U and replace E again by — dS/dt, the Hamilton-Jacobi equation assumes the form

±(\gr*dS\Y+U + ^ = 0 (10)


or

This is the "wave equation " of classical mechanics. In optics the law according to which a scalar quantity gxyy,zj) (e.g., the component of the electric field strength in an electromagnetic wave) is propagated is governed by the wave equation :

where \x = index of refraction, c = velocity of light in empty space. We try a solution of the form

gx,y^t)=f(x,y,zy* (13)

The phase (f> is given by

^ = 2 . ( 4 - ^ = 2 ^ ( 1 - * ) = ^ - , ) (14)

where v = frequency, A = wavelength, / = geometrical path length of optical path.

We shall assume that the amplitude function f(x,y,z) changes so slowly that we can assume it constant. Substituting Eq. (13) into Eq. (12) we obtain for

(15)

36 326 „ But -£=-27w, -r£ = 0

We shall assume also that the index of refraction changes very slowly (grad fji = 0) and that the rays are nearly parallel (div = 0, where S0 is a unit tangent vector along the optical path.) Then Eq. (15) reduces to

But this is an equation of the same type as Eq. (11). Schrodinger therefore assumed that S is proportional to </>. Since S has the dimension of an action and tf> is a pure number, Schrodinger put

(f' = 2 J s = = T S * - E t ) ( 1 7 )

since, according to Planck's law,

4 = * 08)

CLASSICAL MECHANICS §1.12

Schrodinger assumed that the behavior of a particle can also be described by a wave equation :

1 d2W

He put Y = if>xyyyz)e~2™Etlh (20)

where ^(xyyyz) = e2niS*lh

and therefore, if this value (20) is substituted into Eq. (19), we obtain

V V = - ? 5 r * (21)

but u2 = 2mE - U)

and thus we obtain the celebrated Schrodinger equation :

V Y + ^ - ( * - E W = 0 (22)

1.12. The Lagrangian and Hamiltonian formalism for continuous systems and fields. So far only systems with a finite (or perhaps denu-merably many) degrees of freedoms have been considered. We shall deal now with a continuous system, such as a fluid or an elastic solid. The ensuing relations could be derived as limiting cases of the corresponding laws for discrete systems. The classical treatment of continuous systems is a very important preliminary step to the quantization of wave fields, the so-called " second quantization."

The dynamical variables of particle mechanics, which are functions of a finite number of degrees of freedom, become now functionals, i.e., functions of infinitely many variables, namely the values of the field variables at each space-time point. We shall denote the field variables by yj&A = l,...,iV). The Cartesian coordinates will be denoted by xly x2y and xs. The yA play now the role of the generalized coordinates.

yA = yA(^x2)xZyt) (1)

Partial derivatives with respect to spatial coordinate, will be denoted by commas :

,, - d y A n \

yA,r =-fr (2) and partial derivatives with respect to time by dots :

dt v, - d y A n \ yA = ~xr (3)

§ 1 . 1 2 CLASSICAL MECHANICS 175

The yA play the role of the generalized velocities q. In the case of the electromagnetic field we have four field variables (N = 4), viz., the three components of the vector potential A and the scalar potential <f).

We assume again that the " equations of motion " are the Euler-Lagrange equations of a variational principle :

8L = 0 (4)

£dV (5)

where dV is an ordinary three-dimensional volume element. The variations are taken only in the interior of the " world-domain " t, V, and the independent variables x, t are not varied. We shall assume that £ , (the " Lagrangian density " ) , is a function of the j % yA r, a.ndyA only and not of x and t, or any higher derivatives of the yA :

£=£(yA>yA,r,yA) (6)

The resulting Euler-Lagrange equations are

fyA S I tyXr J 8 t d$A ~ ^ These equations are called the field equations. One calls

8L y 3 3£ 8£ &L tyA r 3 x r 3yA,r ~~ fyA ~~ djA

the variational derivative of £ with respect to yA or the functional derivatives of L with respect to yA. The energy-stress tensor ta

Q is defined by

where p, a = 1, 2, 3, 4. The index 4 refers to the time coordinate t. On account of the field equations (7) the tensor ta

Q obeys the conservation laws :

J v , f = 0 (9) Q=l

These are four ordinary divergency relations, expressing the conservation of linear momentum (a = 1, 2, 3) and energy (a = 4).

In analogy to particle mechanics the momentum densities 11^ are defined to be

TT d£ /,^X

176 CLASSICAL MECHANICS § 1.12

H is defined the same way as in particle mechanics :

N

X •tV,

M(UA9yA9yA9r) = X n ^ -£[yA,yA,r9yA(yA,yA,rJlA)] (12)

In this expression we assume that the velocities yA have been expressed in terms of the momenta. The canonical equations are

8H SH •

yA = m-A^m2 (13)

n^, = - — + — = - — ( r = 1,2,3) (14)

If F and G are any two dynamical functionals, i.e.,

F = n f f ( y A j A , r J l A W (15)

and G=Mg(yA,yA>rnA)dV (16)

then the Poisson bracket of F and G is defined by

In particular, the time rate of change of F is given by

For example, in the electromagnetic field in free space, the field variables yA are Al9 A2, Az, tf> (i.e., the vector and the scalar potential). Maxwell's equations

curl E + — - = 0 ; c u r l J ? - — — = 0

d i v £ = 0 ; d i v # = 0 I

are the Euler-Lagrange equations belonging to the Lagrangian density

r = ( l i ¥ + ^ l ) 2 - ^ ( i c u r i ^ ) 2 <2°)

The next step is to define the Hamiltonian density H. The Hamiltonian H itself is then

H=ltfHdV (11)


The conjugate momentum density to Ax is II 1 ; where

The conjugate momentum to <f> vanishes identically since 3<f>/dt does not occur in the Hamiltonian. The Hamiltonian density is

3A 1 H= n •-£-—£ = 2rrc* | n |2 + f -(| curl ^ |)2 - ;II • (22)

The canonical equations are

^ = Aircm - c grad </> (23)

and = — - i - curl curl 4 (24) Ot 47T

These equations give us all of Maxwell equations except the third

d i v £ = = 0

We must require that div E = 0 at some instant of time. But then the canonical equations give us automatically the result that this restriction is maintained at all times.

Bibliography

1. CORBEN, H. C. and STEHLE, P., Classical Mechanics, John Wiley & Sons, Inc., New York, 1950. An excellent account of the mechanics of continuous systems and fields.

2. GOLDSTEIN , H., Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass., 1950.

3. LANCZOS , C , The Variational Principles of Mechanics, University of Toronto Press, Toronto, 1949.

4. SYNGE , J. L. and GRIFFITH , B. A., Principles of Mechanics, 2d ed., McGraw-Hill Book Company, Inc., New York, 1949.

5. WHITTAKER, E . T., A Treatise on the Analytical Dynamics of Rigid Bodies, 4th ed., Cambridge University Press, London, 1937. The standard work.

Chapter 6

SPECIAL THEORY OF RELATIVITY B Y H E N R Y Z A T Z K I S

Assistant Professor of Mathematics

Newark College of Engineering

1. The Kinematics of the Space-Time Continuum

1.1. The Minkowski 4 4 world. " Ordinary three-dimensional space plus the time form the four-dimensional " world." A " world point " is an ordinary point at a certain time. Its four coordinates are the Cartesian coordinates, x, y, z and the time t, which will also be denoted by x\ x2, #3

and # 4 . An " e v e n t " is a physical occurrence at a certain world point. In what follows, the Einstein summation convention will be used : Whenever an index appears twice, as an upper (contravariant) and a lower (covariant) index, it is to be summed over. If the index is a lower-case Greek letter, the summation extends from 1 to 4 ; if it is an italic lower-case letter, the summation extends from 1 to 3. (Generally, the indices 1, 2, and 3 will refer to the spatial dimensions and the index 4 to the timelike dimension of the continuum.) Examples are

aTrbr = aT1b1 + ax% + ar9b3

The two most fundamental invariants of the special theory of relativity are the magnitude c of the speed of light in vacuo and the four-dimensional "d i s tance " r 1 2 of any two world points ( 1, 1, 1, 1) and (x2,y2yz2yt2) defined by

V = (h - h f - 1 [(*, - x,f + ( y 2 - y i f + (*, -

(1) = ( A 0 » - - ^ [ ( A * ) » + (Ay)» + (A*)»] .

All the kinematic properties of the theory are a consequence of these invariants.

178

§ 1 . 2 SPECIAL THEORY OF RELATIVITY 179

1.2. The Lorentz transformation. We introduce a covariant " metric tensor " rj^ and its corresponding contravariant tensor rj^, defined by

= 8 t

where

and

\ i I 0

and

0,

0,

O,

O, O, O,

if if

O,

O,

.0,

fJ,=V

O,

O,

O,

(1)

(2)

O,

0, O,

O, O,

O,

0

0

+ 1 J

0 0 0

I-1

(3)

In terms of the metric tensor, the " world distance " o r l 2 becomes

world interval "

(4) When we go from one reference system S with the coordinates xx to a new system S with the coordinates xx the only admissible systems are those resulting from a nonsingular, linear transformation that leaves T 1 2

2 invariant. Since only coordinate differences and not the coordinates themselves are involved, the transformations may be homogeneous or inhomogeneous in the coordinates. Usually, only homogeneous transformations are considered: the " Lorentz transformation." Think of them as rotations in the four-dimensional continuum (the inhomogeneous transformations would correspond to rotations plus translations. The transformation equations are

• A V ) (5)

xx = y:

where the y'*. denote the coefficients of the inverse transformation and are solutions of the equations

K ( 6 )

The y\'x themselves are the solutions of the equations

(7)

180 SPECIAL THEORY OF RELATIVITY §1.2

A spatial rotation in three-dimensional space is a special case, characterized by yr.\ = 0 and y\\ .= 1.° Another special case, the most important one in the subsequent discussion, is that one giving the relations between a coordinate system 2 at rest and another system 2 moving in the direction of the positive x axis with the constant speed v and whose origin O coincides at the time t = 0 with the origin O of 2 . The transformation coefficients y\\ are in this case.

1 o, o, — z; V i

o, o, V i - ©a/r2

o, 1, 0, 0 o, 0, 1, 0

- v lc2

0, rv 1 V i - v^/c2' 0, U, V i - v2jc2 _

1 o, 0, V

V i - v2/c2' o, 0, V i - v2\c2

o, 1, o, 0 0, o, 1, 0

vie2

o, 0, 1

V i - v2\c2' o, 0, V i — v2jcl_

(8)

(9)

The relations between the two coordinate systems are *, * ' x — vi

x*= t =

--y

-- Z •

v2jc2 VT

-y

-- z

t — vx/c2

x + vt VT

--y *

= z t + vxjc2

Vl - v2]?

The ratio vjc will be denoted by /?.

> (io)

> (11)

§1.3 SPECIAL THEORY OF RELATIVITY 181

The Lorentz transformations form a group, i.e., if the system 2 is related to the system 2 by a Lorentz transformation, and if the system 2 is related to 2 by another Lorentz transformation, then there exists a third Lorentz transformation which relates 2 directly to 2 .

Lorentz's transformation laws [Proceedings of the Academy of Sciences, Amsterdam, 6, 809 (1904)] are however, not completely equivalent to Einstein's. His formulas (4) and (5) read :

Therefore x2 — c2t2 is not an invariant, since

x2-c2i2 = x2-c2t2 + 2Pxct + c2p2t2.

1.3. Kinematic consequences of the Lorentz transformation. a. Relativity of simultaneity. Order of events. If an event occurs in 2

at the points xx and x2 at the times tx and t2> then

* tx — vxjc2 * _ t2 — vx2\c2

V T ^ F ' h ~ v i = w ( 1 )

and therefore

i ; _ • h-h-(P/*2)(*2-*i) m

If t2 = tly we obtain

Thus two events that are simultaneous in 2 are no longer simultaneous in 2 . If the two events in 2 can be connected by a signal whose speed w is less than the speed of light, i.e., if

^ = f i s r o > c (4)


d V (11)

Hence _ ' = ' 3 _ < x = c ( 1 2d^ = n )

Suppose now that AB is perpendicular to the velocity v. In this case the

then * « - * _ = (*_-*_) / t ' (5) V I — /£

In other words, £2 ~~ *i will always be positive if t2 —1± is positive. The temporal order of events is not destroyed by a Lorentz transformation.

b. The time dilatation. If a clock, e.g., an atomic oscillator, vibrates with the period T at a fixed point a moving observer will observe the period

c. The Lorentz contraction. The spatial distance between two world points P1 (x^y^z^t) and P2 '= $2>y2yz2tt) is

L = V(x2 - x,f + (y2-y,f + ( i , - JJ" j ( ? )

= V f e - ^ ) 2 ( l - i 8 ) 2 + ( y 2 - ^ ) 2 + ( ^ - ^ ) 2 ) The linear dimensions in the direction of motion appear contracted in the

ratio 1: 'sj 1 — jS2 to a moving observer. Distances orthogonal to the direction of motion remain unchanged. This contraction explains the negative outcome of the Michelson-Morley experiment, historically the starting point for the creation of the theory of relativity. Suppose that an observer on a material system, moving through space with a velocity v, makes observations on the velocity of light, and determines the time taken by light to pass from a point A to a point B at a distance d from A and to be reflected back to A. Let the velocity v of his system be in the direction from A to B. Let the light start from A at time tlf arrive at B at time t2, and get back to A at £3. Then if the light travels through space with velocity cy we have

d + v(t2-t1) = c(t2-t1) (8)

d-v(tz-t2) = c(tB-t2) ( 9 )

d or h — h = (10)

• c — v v 7

§ 1 . 3 SPECIAL THEORY OF RELATIVITY 183

1 + vujc2

1 + vujc2

usyr^j2

1 + vujc2

1 + vujc2 v '

(18)

Applications. (1) Assume that a light ray is traveling in the j> direction, ux = M 3 = 0 ; w2 = £. Then ux = — v; w2 = ^Vl — jS2; % = 0. The angle a which the ray includes with the y axis is given by

— " i — " ( I + f + - ) < 1 9 )

For small values of j8, the absolute value of oc becomes jS (angle of aberration).

( 2 ) Let N be the refractive index of a substance having the velocity v in the x direction. The speed of light in this substance is u = cjN. Consequently the speed w of a light ray traveling in the x direction is

» = « i = r+i%? = jf + "(l - w) + - ( 2 0 ) the factor (1 — l/N2) is called the Fresnel-drag coefficient.

distance traversed by the light signal, as it travels from A to B and back to A again is

/ 2 ~ ' ~ !

D 2 + T (V - V ) 2 = < V - V ) = (13)

Therefore A * ' = — , 2 * * (14) cVl-p2

The two time intervals A* and A/ ' are not the same. However, because of the Lorentz contraction, the required time A* is shortened, and becomes

It was experimentally observed that A* — A*'.

d. Velocity addition theorem. If the velocity of a material particle has the Cartesian components uly u2y and us in S , its components in S are given by

(16)

184 SPECIAL THEORY OF RELATIVITY § 13

e. Doppler effect and general formula for the aberration. If a plane light *

wave in vacuum travels along a direction which forms an angle 0 with the x axis and has a frequency v in the system 2 , an observer in the system 2 will observe the frequency v and the angle 0, where

(1 + B cos 0) * v — -——===-=— V

V i - j S 2

cos 0 + ft 1 + j3 cos 0

cos 0 =

(21;

(22^

sin 0 = Vl - )82 sin (9 1 + j8 cos 0

(23)

f. Reflection at moving mirror. Let a mirror have the speed u and a plane wave be reflected by it (Fig. 1). If ^ is the angle of incidence, and I/J2 the angle of reflection,

ran i -^T— I (24) (?)

c + u

If v ± is the frequency of the incident wave and v 2 that of the reflected wave,

(25) v2 _ sin i/rj

FIGURE 1


If the mirror moves opposite to the incident wave, we note that ifj2 < iff±

and v2 < vx. These formulas played an important part in Planck's derivation of black-body radiation and in the thermodynamics of cavity radiation.

g. Graphical representation of the Lorentz transformation. We project the four-dimensional world upon the x,t plane (Fig. 2) and choose as abscissa

FIGURE 2

not the time t but £ 4 = ct and as ordinate x == gv The relations between * *

x' = f 1/ and ct = £ 4 , and £ x and £ 4 are given by

| 4 = — i£i sin a + £ 4 cos ac (27)

or if we put ^ = and *£ 4 = ?y4,

— Vi C O S A ~"" ^ 4 s m a (28)

T?4 = = % S M A + ^ 4 c o s a (29)

where i tan oc = vjc.


Introducing the real angle y, defined by y = /a, we can rewrite these formulas in the form

\\ = l i cosh y — £ 4 sinh y (30)

h = — £i sinh y + £ 4 cosh y (31) *

The expression c tanh y = z; is the speed with which the reference frame 2 moves with respect to the reference frame 2 .

A uniform motion of a particle, passing through the origin of either coordinate system at the time zero is represented by a straight line through the origin. Its slope is given by tan <j> = w\c where w is the speed with respect to the system 2 . Since the speed cannot exceed the speed of light, the two 45° lines passing through the origin represent the motion of a light signal. In the Minkowski world the light signals reaching the origin at the time zero form the surface of a three-dimensional cone with its vertex at the origin. Likewise the light signals leaving O at the time zero form another cone with the vertex at O. This double cone is called the " light cone." Any physical event either in the past or in the future that can be linked to an event happening at the origin at the time zero must take place at a world point located in the interior or on the boundary of the light cone. The left-hand cone represents the past and the right-hand cone the future. All other events are located in the " present," since they can neither have caused nor be the consequence of the event at the origin at time zero. In the sense of the Minkowski metric the "d i s tance" between two points Ptfî) and Ptft'ti) is diPJ>t) - + V f o ' - ~ (t7=W. where it is assumed that both points lie within the same part of the light cone, i.e., both points lie in the past or in the future. It is an immediate consequence of this definition that

d(OPr) + d(OP2) 52 d(PJP2) (32)

In other words, the usual triangle-inequality is just reversed. The straight-line connection between two points is not the shortest, but the longest connection.

2* Dynamics

2*1. Conservation laws. Classical mechanics knows two independent conservation laws, the conservation of a vector ps, the linear momentum, and the conservation of a scalar, the energy E. Relativistic mechanics knows only one conservation law, the conservation of a world vector Pe.

§ 2.1 SPECIAL THEORY OF RELATIVITY 187

1 2 n

Pe = constant (p = 1,2,3,4) (1) * - i *

— ^ = = > — \ (2)

The corresponding covariant vector is

( — mus E\

_ u*lc29 c

2 ) . fe y I _ „2^2 * The relativistic kinetic energy T is defined by

T = — l L = - m * ( 5 ) Vl - U2\c2

which for small velocities becomes the classical value \mu2. Since the vector Pe is conserved, its magnitude, and hence also the square of its magnitude are also conserved. Thus

1 rp2 JL constant (6)

When the expression for E and/>s are substituted into Eq, (6), one finds that

P?PQ = rn2 (7)

If the rest-mass is zero, e.g., in the case of a photon, we obtain the result that

X w 4 (8)

In words : the magnitude of the linear momentum is p = Ejc. On using the Planck relation that E = hv (v is the frequency of thephoton), we obtain

T o illustrate the application to the Compton effect, let us assume an electron, of rest-mass m, is initially at rest and struck by a photon of frequency v.

More precisely, it states that if there are n particles present with rest masses m> m, w, then in the absence of external forces four laws of inertia hold :

188 SPECIAL THEORY OF RELATIVITY § 2.2

The scattered photon has the frequency v' and forms an angle 8 with the direction of the incident photon. The electron acquires the velocity v in a direction making an angle cf> with the direction of the incident photon. The conservation law for the energy states that

h - i ' + vfTW (10)

The conservation law for the momentum states that

hv hv' a mv — = — cos V -\ —_=__=--=_ cos 6 (11) C C y 1 _ V2/C2 r V

i ^ hv' . n mv , and 0 = sin U _ ——- sin tp (12)

c V l — v2jc2

By eliminating </>, we obtain

mv-v')~^\ -cos8)vv' = 0 (13)

On writing 1 — cos 8 — 2 sin2 8/2; v — cjX; v' = c/X' we finally obtain

A ' - A = — s i n 2 ^ - (14) mc 2 .

where hjmc is usually called the Compton wavelength.

2.2. D y n a m i c s o f a free mass point . The dynamics of a free mass point of rest mass m follows from a variational principle :

i = \ l L x ^ y u = ( i )

8 7 = 0

The variation is to be performed only in the interior, the end points are kept fixed. The Lagrangian function L is given by

L = —mc2 V l - v2jc2 (2)

The corresponding Euler-Lagrange equations are


The momenta are, as usual, defined by _ 3L mxs

P s ^ d ¥ = ^/Tzr^p ' (4) Equations (2.2) and (2.3) can be brought into the canonical form. We introduce the Hamiltonian funtion H, defined by

tnc2 / P2

H = —L + pjcs = —— = mc2 \ 1 + -^ -^ , .

(P2 = Pi* + P2* + P32)

The canonical equations are .S_3H _ pjm

(6)

2 3 . Relativistic force* The force fs acting on an accelerated mass point can be defined, just as in classical mechanics, as the time rate change of linear momentum :

(i) w ( v T = y - K r - M ' - £ ) + ~ ] (v2 = x* + y2 + z2) /

This force is in general not parallel to the acceleration. It is parallel only when the acceleration is either parallel or perpendicular to the velocity. When it is parallel, Eq. (1) takes the form

7 . = 1 - ^ ™s (2)

If the acceleration is orthogonal to the velocity, Eq. (1) takes the form

fs= ? ) rnx8 (3)

The coefficients (1 —v2jc2Yzl2m and (1 — v2jc2)~ll2m are called the " longitudinal mass " and the " transversal mass " in the older literature.

2.4. Relativistic electrodynamics. We introduce the covariant tensor (f> = (AS) — c<f>),

As — vector potential, </> — scalar potential

190 SPECIAL THEORY OF RELATIVITY §2 .4

the world current-vector = (Is>p)>

Is = current density, p = charge density

and the world force density Fm = (ksy A/c2),

ks = ordinary force density, A = power density

Comma means differentiation, e.g., tf>Q a = dtf>Jdxa and

E = (EvE2yEz) — electric field strength

D = (DlyD2yD3) = electric displacement vector

H = (HlyH2yH3) = magnetic field strength

B — (BlyB2tB3) = magnetic displacement vector

We finally introduce two antisymmetric tensors ® e a and Y e < T :

( f l W J W O a ) = -B, -Blt-B2)

( < D 1 4 , < D 2 4 , < D 3 4 ) = (-cEv-cEt-cEJ

( T ^ Y ^ Y 4 3 ) = (-c»D 1 , -<»D 1 , -c»D,)

The fundamental equations of electrodynamics are

s\x = 0 or div 1+^ = 0 (1)

^ , , = 0 or d i v ^ + l ^ = 0 (2)

1 dA <f>A.Q—<l>Q.x=+®xQ or B=curlA; E = - grad <j> - y y - ( 3 )

1 dl?

<H,« + < 1 W + «X$, or

I A = / £ ( 7 )

In vacuum E = D and H^= B and therefore = In this case we can eliminate 0 A p by combining Eq. (3) and Eq. (5) and utilizing (2) we obtain

\J(f>Q = 47TCSe (8)

where • is the wave operator or the " D'Alembertian "

d2 , e2 l a2

n dx2 1 ay2 ' dz2 c2 dt2

If we put

we obtain 4TTCQ^ = • ZEK (9)

where Z e A corresponds to the Hertz vector in electrodynamics.

2.5. Gauge invariance. The Maxwell equations (1), (4) and (5) of the preceding paragraph remain unaltered if the world vector (j>Q is changed into a new world vector fQ by adding to it an arbitrary four-dimensional gradient field ifjyQ :

$e=<f>Q + *l>>Q , (1) Such a transformation is called gauge transformation. The condition (2.4-2) is not gauge-invariant. But since ifj is arbitrary and does not affect the Maxwell equations, we are permitted to make this choice. The advantage of having the " Lorentz gauge condition " (2.4-2) satisfied is that we can solve for the highest time derivatives of the potentials and thus obtain a system of differential equations of the Cauchy-Kowalewski type. In such a system the initial conditions on a function guarantee its unique continuation into the future (and its past).

If one works with the field strengths E and H themselves (which are the only physically observable quantities), rather than with the potentials, then Maxwell's equations are automatically of the Cauchy-Kowalewski type with respect to the time derivatives, regardless of any gauge condition. We introduce potentials primarily for mathematical convenience.

192 SPECIAL THEORY OF R E L A T I V I T Y § 2.6

£ 3 = . 7 ^ = ^ + S3 = . 7 = = (B, - i8£2)

1

V i -1

Vi - £ 2

1 V i - j82

1

V i - |82

A> = . 7 f ^ ( D 3 + Wz) H, = — J — ( f f , - 0Z)2)

V i - j82

1

V i -

1

V i -1

Vl - j S 2

(1)

J

Here j8 = v/c, where v is the velocity of S with respect to 2 and ^ = (s>,0,0). The total charge is an invariant. If the world current vector sQ is purely convective, and if the charge has the velocity u in the system 2 and u in 2 , then

Pui = P Vl - j82

pu2 = pu2

puz = puz

* _ 1 — vux\c2

p-p~VT^W If the charge is at rest in 2 , i.e., if ux — u2 — uz = 0, then

* _ P

(2)

2.7. Electrodynamics in moving, isotropic ponderable media (Minkowski's equation). The relations between 0 - A and WQX must be of such a nature that, in a reference system at rest with respect to the medium, the relations must go over into

D = e £ (1) B = ( l H ( 2 ) Ic = oE (3)

2.6. Transformation laws for the field strengths

Et = Ex Bi =B1


T J Q = ( U S - 1 ) W l —uHc* cVl -u2lc2J *jc* cVl - u2/c

We shall denote the corresponding world vector of Ic by le. Therefore

l* = s*-U°- (sxUi) (3)

becomes / = oE (4)

We introduce two new world vectors

eQ = <!>*XUX and dQ =YQXUX

Then dQ = eeQ ( 5 )

is an invariant relation which reduces to D — eE in a system at rest with respect to the medium. We also introduce two antisymmetric tensors

bQXx = < J ) 0 A C / T + <S)XrUQ + <SXQUX

and NEXR =YeXUr+YXrUe+YreUx

Then bQxX = fihexX (6)

is another invariant relation, which reduces to B = \jJEL in a system with respect to the medium. We introduce certain abbreviations :

E' Ê+-^-(ux B)

D' ^D + -^(ux H)

H^H--(uxD) c v

B' ^ B — - ( * * £ ) c v

(electromotive force)

(magnetomotive force)

Then the Minkowski equations read

D' = eE' (7)

B'=ilH' (8)

The generalization of Ohm's law is

€ = dielectric constant, /x — magnetic permeability, cr = electric conductivity, / c , the " metallic current density, is obtained from the total current density se through subtraction of the convective current density Ue • (sx Ux), where

l e = <J£Q (9)


In vectorial form this means

1 + c2-u2 Vl~u*lc* ( j

If the current is purely convective, then / = pu and se = Ue • (sAL7A), and the Lorentz-force expression becomes

* = £< iwa ( I D The foregoing equations are to be regarded as representative of various

competing tensors and not as completely verified formulas.

2.8. Field of a uniformly moving point charge in empty space. Force between point charges moving with the same constant velocity. We shall assume a point charge e muving with+the constant speed v in the positive x direction. In a reference system 2 , chosen so that the charge is at rest with respect to it, and permanently at its origin, we have

*s

Ds = Es=s~ B s = F s = 0 (1)

Using the transformation equations of F, we obtain

E1 = EV i / 1 = 0 * *

A « - v T 3 7 P ' H*-~VT^W 3 \ (2)

W — E* Tl — A h Vi - p2 Vi-p2 )

Since r2 = x2 -\- y2 + %2 and x = y—y, st = z Vl - ] 8 2 '

we obtain Et

E9 =

£ X — vt Hi = 0 V i - p*

Hi = 0

e y H2

V i -H2

Vl - P2 Ra

e z H3

_ efi . y

P2

H3 Vl - p2 R3

(X — Vt)2 0 | 2 # 2 = j _ ^ + J>2 + ^

( 3 )

§2.8 SPECIAL THEORY O F RELATIVITY 195

These expressions show that the electromagnetic field is carried rigidly along by the charge (" convective field " ) . For simplicity, consider the field at the time zero, when the charge passes through the origin. Then

and

R 2 = Y^rp+y2 + z*

£, = , • —, HT = 0 £ X

V i - |82

8 y Vi - 0 2

s Z

Vi - |82

E2= , " HT= — \ (4)

Vi - £2 ie3

The magnetic field lies entirely in the " equatorial plane," i.e., the plane orthogonal to the direction of motion. Let us compute the electric field strength at a distance s from the origin. If the point of observation lies on the x axis, R = s]^/1 — /J2, we obtain

£ i = £ ( 1 ^ 2 ) ; E2 = E3 = O (5)

If the point lies on the y axis at a distance S from the origin, we obtain 2? = S and

£ 2 = / £ x = £ 3 = 0 (6)

As the speed increases to the speed of light, in the first case E1 = E2 = EZ = 0 and in the second case E2 = 00; EX = E3 = 0. At high speeds the electric field tends more and more to crowd in the equatorial plane. The energy density becomes zero outside and infinite inside the equatorial plane in such a manner that the total energy of the field becomes infinite, and therefore also the electromagnetic mass of the charge becomes infinite. The Lorentz force F exerted on a second charge e! which moves with the same velocity ir, is

F=e'\E + ±[uxH]\ (7)

and can be written F = grad 0,

where 0 = (1 - j S 2 ) £ (8)

Here ift is called the "convective potential."

196 SPECIAL T H E O R Y OF RELATIVITY §2.9

The surfaces of constant convective potential are

R2 = x2 + (I - ft2) (y2 + z2) = constant (9)

They are ellipsoids of revolution with the charge at their center. The direction of motion is the axis of revolution and the principal axis is smaller than the two minor axes. These ellipsoids are called the " Heaviside ellipsoids."

If the charge e' lies on the x axis at a distance s from e, the magnitude of the force exerted on it, is

F = ^ \ - ? ) (10)

If the charge is located on the y axis at a distance s from the charge e, the force is

F = £VT=P (11) In either case the force between the two charges tends to zero as their velocity approaches the velocity of light.

2.9. The stress energy tensor and its relation to the conservation laws. We introduce the tensor Mea, defined by

1 ~4~ toxfi*" - \ ( O ^ T ^ + ®aXYw)r]^ (1)

47TCA

Q^Y^^lc^B-H-E-D) (2)

Mrs = i - 1 8 r s ( B H + DE)- BrHs - DrEs (3)

M" = ^rc(Dx B + ExH)x (4)

M" = ^(E-D + H-B) ( 5 )

The Mrs are the components of the Maxwell stress tensor; c2M*s are the components of the Poynting vector; c2Mu is the energy density of the electromagnetic field. The energy stress tensor obeys the four conservation laws

M*\i = 0 (6)

If in addition to the electrodynamic stresses MqX any other stresses PeX

of nonelectrical origin are present, the conservation laws take on the form

T>\x = 0 (7)

where I** = P*A + MeX == " total stress energy tensor."


As a special example assume matter of constant rest density cr moving with uniform velocity v along the x axis. Assume furthermore that the mechanical stresses are given by the tensor trs. Then

t" \ a V 2 a 4 - vH"Ic* P" — ' U . p 2 2 _ _ ^22 . p 3 3 _ _ ^33 . p 4 4 _ u \ u /<' 1 -v2/c2y ' ' 1 ~ ^ 2 / c 2 '

Vi - v2!c2' Vi - »7* 2 ' i - ^ 2 ' V t12 V t l z

P 2 3 _ /23 . D24 _ _ . _ . P 3 4 =

L L > Z 1 - . 2 / - 2 » X ~2 ^2 I _ V2jc2 > C2 I _ ^ 2 / r 2

In the case of a perfect fluid of pressure p, we have trs = ^ § r s . ^,12 = ^ 1 3 = ^ 2 4 = ^ 3 4 ^ ^ 2 3 = Q

u _ _ p + civ2 . q + ^ P ~ l - z ; 2 / < : 2 ' P P ~ P ' ? - 1 _ _ ^ 2 ^ ( 9 )

, ,14 = RV+.VpJc* p^ = 1 - <y2/c2

3. Miscellaneous Applications

3.1. The ponderomotive equation. The Lorentz-force on a particle of rest-mass m and charge s can be derived as the Euler-Lagrange equation of the variational principle

/ = ^ ' L A = J ^ j - « c » . ^ l - ^ - ^ + 4 - « ^ f JA (87 = 0) (1)

In this variation the end points are kept fixed. The Euler-Lagrange equations belonging to Eq. (1) are

d_ dt ( v r = y = e E s + f = - = e t c - ( 2 )

The " relativistic momenta " ps are defined by

The velocities can be expressed in terms of the momenta:

(4)

,198 SPECIAL THEORY OF RELATIVITY §3.2

The Hamiltonian function H, defined by H = psus — L is given by

H = mc2 1 + — 9 * .'. + e<£ m2£2 J T

e<f> ( 5 )

(6)

The canonical equations of motion are

Equations (8) are equivalent to Eq. (2).

3.2. Application to electron optics. We consider the motion of a charged particle in a combined stationary electric and magnetic field.

£ = - g r a d < £ ; H=curlA (1)

The Lagrangian is

L = m*(l-<Jl-^-*l> + ^(u'A) (2)

The variational principle 8 J^2 < dt = 0 yields

The total rate of change of the potential energy <f> of the particle along its path is given by

| = | + . - g ^ ,4 ,

Since we assume a stationary electromagnetic field,

I - O and thus « • grad <f> = ^ (6)

Therefore the Lagrangian L can be written in terms of the momenta and is given by


Multiplying both sides of Eq. (3) scalarly by n, we obtain

dt (A /1 - u2/c2) 3dt

or •dt\< Vl - u2/c2

Or introducing the relativistic kinetic energy

1=

<f>)=0

mc'

V l - u2jc2

Eq. (8) can also be written

d / mc2

— mc 2 + 8<f^ = 0 LVi— u2/c2

And thus one obtains one integral of the motion, the energy E,

'2 + ecj> = E V l - u2/c2

8<f) = E + mc2

V l - u2/c2

This equation enables us to rewrite the Lagrangian in the form

L = E-e$)\+ A\-^+Û-A)-E

Since u = dsjdt, we obtain from Eq. (10)

and

Ldt =

ds__ V(E + 2mc2 - e<f>) (E - &£) dt E + mc2 — s(j>

V ( E - e ^ ) (E + 2mc2 - e<f>) + y (s • A) j ds - E dt

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

where 5 is a unit vector in the direction of the tangent of the trajectory of the particle : u dt = s ds. The variational principle can be reformulated

§ T2 — V(E -8<j>) (E + 2mc2 - ef) + — (s- A))ds = 0 (15) J Pi c c I

since E is a constant and t is not varied, and therefore

8 r2Edt = 0.


If we call the dimensionless quantity /x

> = i [ V ( E ~ S<l>) E + 2mC* ~ S<l>) + 6(5 ' A ) ] ( 1 6 )

the variational principle is

8 j ' V & = 0 (17)

This is identical with Fermat's principle in geometrical optics. \L is the " index of refraction.'' It depends through </> and A on the position (x,y,z) and through s also on the direction. The index of refraction is thus seen to be both inhomogeneous and unisotropic. Equation (16) is the fundamental equation of electron optics.

3.3. Sommerfeld's theory of the hydrogen fine structure. We shall assume that an electron of rest-mass m and charge e moves in the field of a proton which is assumed at rest. Let r be the distance between electron and proton. The energy E is given by

mc2f . 1 ^ - l ) = E (1) The angular momentum

mr2d

Vl - u2lc2

is also a constant of the motion. According to Sommerfeld's quantization rule

§Iedd = neh (2) The integration must be carried out over a complete cycle of the motion, ne = 1, 2, 3, ... (nd — 0 would imply that the electron penetrates through the nucleus.) Therefore

2TT = ne - h (3) Vl - u2\c2

mr26 = ne • - u2jc2 % = (4)

The radial momentum pr is given by


(8)

Usually nr + w0 = n is called the principal or orbital quantum number, and n6 = k the azimuthal quantum number. Hence

2%H2 n*\k 4 / 1 + 3 - t - T W

Thus the energy depends not only on w, as in the classical theory, but also on k. If the speed of light were infinite, then oc = 0, and E would depend on n only.

4. Spinor Calculus

4*1. Algebraic properties. The tensor formalism does not include all the objects that form relations that are invariant under Lorentz transformations. Besides the world tensors there exist " spinors " whose transformation group is also Lorentz-covariant and whose representation admits an even larger possibility than that of the tensors. In that sense the spinors are more fundamental than the tensors. In order to facilitate the reading of the literature * we shall change slightly our previous notation. The coordinates (x,yyz,ct) will be denoted by (x1,x2,xz

yx*) and the metric tensor will be assumed

TÎI = 77 2 2 = 7 f 3 - 1; 7? 4 4 = - 1

* See especially the paper, " Application of Spinor Analysis to the Maxwell and Dirac Equations," by LAPORTE , O. and UHLENBECK, G. E., Phys. Rev.y 37, 1380 (1931). The following discussion follows this paper closely and the reader will find further discussion and applications in it.

It must again obey the same quantization rule

§prdr = nrh (nr = 0,1,2,...) (6)

If Eq. (4) is substituted into Eq. (6) and the integration is carried out, we obtain

E = mA\--— 1 = ] (7)

L V l +<x2l(nr + V « e2 - a 2 ) J

where a = eij%c is the fine-structure constant. If E is" expanded in a power series of a, we obtain


We shall call any two-dimensional vector g = (gvg2) a spinor of rank one. Its components obey the following linear transformation law

i l = a l l £ l + a 1 2 # 2 ^

h = « 2 l £ l + ^22^2 )

and where the determinant of the transformation coefficients has the value unity

1 (2) a l l a 1 2

«21 a 2 2

the spinor £ will also be denoted by gk(k — 1,2). Spinor indices will always be denoted by lower-case italic letters and always assume the values 1 and 2 only. The coefficients ocrs(r = 1,2; s = 1,2) can be real or complex numbers.

We shall also define a spinor / = ( / , / 2 ) with dotted indices, which obeys the transformation law

/ i = 5 l l / i + Ol2 /2

fi ^21 f'l ^22-^2

(3)

This spinor will also be denoted by fkk = 1 , 2 ) . A bar will always denote the conjugate complex.

Spinors of higher ranks are quantities that transform like products of spinors of first rank, e.g., the spinor akl(kyl = 1,2) shall transform like akbh

or the "mixed" spinor a'kl shall transform like akbt. We raise or lower spin indices in the usual fashion by introducing a metric spinor ekl of second rank, symmetric in its indices, such that

aK — eKOai °\ (4)

(1-1) (5)

j a eik __ M _ , ( * _ / ^ M 1 \ - l 0 / J

( 6 )

Summations will be performed only over undotted or dotted indices, but not over mixed indices.


(?)

This ensures that akbk and akbk are invariants. J&om the above definitions,

aP = -a% ( 8 )

Also, the absolute value of any spinor of odd rank is zero.

axal = Q or almnal™ = 0 (9)

flV« + flW + ^ i = o (10)

All these rules hold of course also for dotted indices. The positions of dotted and undotted indices may be interchanged without changing the spinor, e.g.,

arstâsrt = astr (")' However, the interchange of two dotted or two undotted indices will in general change the spinor unless it possesses certain additional symmetry properties. T o obtain the complex conjugate of any spinor equation replace all dotted indices by dotted ones and vice versa.

4.2. Connection between spinors and world tensors. T o each contra variant world tensor Ae a mixed spinor a-rs can be uniquely associated, and vice versa. This permits us to rewrite each tensor equation as a spinor equation and vice versa. The relations are the following :

^ = A = y K i + * i 2 )

A* = A3 = ±-(ah-aJ (1)

Thus e.g., albl = aSbx + a2b2

and ab% = axbx- +

but no summation for alb-

From the above definitions follow the fundamental relations

a1 = a2; b1 = bi

a2 = -a±; b2 = - b


Conversely, we can also express the spinor components in terms of the vector components

aix = ~ a h = A ± + iA* = A i + A 2

ah = - a 2 1 = A1 - iA* = At - iAt

ah = +a** = A3+ A* = A 3 - At

- a i 2 = - a " = # - A* = AS + A,

) (2).

AA« = - ^ a ^ (3)

The above scheme also permits to introduce spinor differential operators . 8

dx1 ~r dx2

8 . 8 dx1 1 8x2

8 3 dx3 a*4

__ 8 8xz ^ 8x*

(4)

By means of the relations (4) all the usual vector differential operators that occur in vector analysis can be translated into equivalent spinor relations, e.g., let <f>'r$ be the spinor that corresponds to the world vector <f>Q; then

- + ^ ! + ^ ! _f_ _L ^ c o r r e s P o n d s t 0 ~2 d'rs<l>h> and if the iQ ~ 8x 8y 8z c 8t ordinary scalar S corresponds to the

spinor scalar s, 32 c 1 a 2 e

corresponds to — ^ - 8^t8rts

4 3 . Transformation laws for mixed spinors of second rank. Relation between spinor and Lorentz transformations. Since mixed spinors of rank two are the fundamental" building stones " for world tensors, we shall write out their transformation laws explicity. The symbol R(oc) shall denote the real part of oc and I(oc) shall denote the imaginary part of a. Thus if a = j8 + iy, where /? and y are real numbers,

R(ot) = P and 1(a) =y

(1)

§ 4.3 SPECIAL THEO RY OF RELATIVITY 205

y (2)

r. - y ( k n | 2 + k 2 2 | 2 - k l 2 | 2 - k 2 l | 2 )

rf;=4-(i-2ii 2+i«22i 2-i«iii 2-i-i212)

The absolute value of a will be denoted by | a | = + Vj8 2 + y 2 . Let g-rs be a mixed spinor of rank two, and g-rs the corresponding spinor

in the new coordinate system. Then

i l l - ' I a i l fel + ^ l i a i 2 & 2 + 5 1 2 a i l & + I a ! 2 fe2 ^

111 = * l l « 8 l f i l + ' « U ° W i a + " u f t l & l + 5 12 a 22^22

* _ _ -

#21 ~ ^2 i a i l f t l ^21^12^ 12 + a 2 2 a i l ^ 2 1 a 22 a i2^22

i 2 2 = I a 2 1 fel + <*2ia22&2 + ^22a21^21 + I <*22 fe2

The relations between the ocrs and the coefficients y\'x of the Lorentz transformation follow :

y\ = ^ ( « i i a2 2 ) + ^ ( ^ 1 2 a 2 l )

y ? i = 7 ( ^ n a 2 2 ) +

r?i = A i ) - ^(^12^22)

r?i = - ^ K i 52 1 ) - # ( a 1 2 a 2 2 )

= 7 ( « 2 2 a i l ) + 7 ( 5 2 i a i 2 )

r?2 = ^ ( ^ l i a 2 2 ) - ^(<*12 a 2l)

r?2 = 7 ( ^ 2 i a i l ) + / ( ( * 12 a 22 )

7% = A ^ l l ^ l ) + / ( ^ 1 2 a 2 2 )

r ! 3 ' = ^ ( « i i a i 2 ) - ^ ( ^ i 0 ^ )

r?3 = 7 ( ^ l i a i 2 ) + 7 ( ^ 2 2 a 2 l )


- -R(au51 2) - i ? ^ !

y?I = - 1 ( 0 ^ ) +/(oâ,.

y ( | a l a . | 2 + | a 2 2 | 2 - I « u I 2 - <*« I2)

y ( l « u l 2 + l a i * l 2 + 1 « 2 1 |2 + <*22 I2) -The y\\ are quadratic in the ocrs. To each Lorentz transformation belong

two spinor transformations which differ only in sign. But to each spinor transformation belongs only one Lorentz transformation. In this sense the spinors are more fundamental than the tensors.

Two special examples will illustrate the meaning of Eq. (2).

an = oc22 = e*oi*; oc12 = oc21 = 0 (3) This yields

r1.! = r?2 = c o s 6

y1.2 = - y 2 . i = s i n d

All other y are zero.

This corresponds to a purely spatial rotation about the z axis through an angle 6.

ocn = e6^; oc22 = e-et2; oc12 = a21 = 0 (4) This yields

y.i = y.8 = 1

y?; = - y ! ; = s i n h 0

y?; = y?; = cosh e *

All other y are zero. This corresponds to a coordinate system S moving in the positive z direction with the velocity v> where tan id = iv/c.

4.4. Dual tensors. We assume the existence of an antisymmetric tensor F*P whose components are all assumed real. The dual of FAP, denoted by F*P, is defined by

F* = rf«ne*FQO (1)


A where FQa is defined by v

£ C a = y W * T (2)

Here hQaXr is either + 1 or —1, depending on whether the indices paXr are an even or odd permutation of the array (1234). If any two or more indices are the same, then 8eaXr = 0. From these definitions, we get

F12 == iFM

F2* = iFu

F31 = iFM

The same relations hold true also for a covariant antisymmetric tensor FQa

A and its dual tensor FQa. A tensor Fâ is self-dual if

A

From a given antisymmetric tensor FQG one can always construct a self-dual tensor GQa by forming

A QQO = fQa _|_ pQa

Whereas an antisymmetric tensor possesses six linearly independent components, a self-dual tensor possesses only three linearly independent components. They can most conveniently be expressed in terms of a three-dimensional, complex-valued vector k•== (klyk2,kz). The components of GQa are then

( G 1 4 , G 2 4 , G 3 4 ) =• -ik^-ik2-ikz)

( G 2 3 , G 3 1 , G 1 2 ) = ( M » * a )

We may write the vector k as a complex sum of two real three-dimensional vectors a and b, i.e.,

k = a + ib (5)

Then ( G 1 4 , G 2 4 , G 3 4 ) = (bx - iavb2 - ia2,bz - iaz)

( G 2 3 , G 3 1 , G 1 2 ) = ax + ibva2 + ib2faz + ibz)

It is possible to associate with a self-dual tensor Gea a symmetric spinor grs

of rank two which also possesses only three linearly independent components, viz.,

p2A=.îpSl > ( 3 )

208 SPECIAL T H E O R Y OF RELATIVITY § 4.5

4.5. Electrodynamics of empty space in spinor form. We construct a self-dual tensor GQa, where

The antisymmetric tensors and FQa are defined by

(la)

pea .

0 # 3 - t f , 0 Ht - E t

0 £ 3

0

and F^0 = — »

0 2?3

0 Et Hx

E i H%

0 Hz

0

(lb)

In our present notation, however, we set c = 1. Therefore the vector A: is in this case k=H—iE (2)

The corresponding spinor has the components

5ii = 2 [ H , + . . £ 1 + t ( H 1 - ^ ]

^ 2 = 2 [ H 2 - £ 1 - ^ 1 + J E ' 2 ) ] (3)

We also introduce a current-density spinor s'rs corresponding to the contravariant current density world-vector

x P P P s = y ®i. y «2» y »s. p. v l z ->

% = y ws + P ' u ^ y ' i + y ' i (4)

•»» + P

Su = 2(k2 - ikj) = 2[a2 - b x - i(b2 + aj\

g22 = 2(k2 + ikj = 2[aa + *i - *'(*a - *i)]

Si* = #21 = = 2[£3 + z«3]

« i = in = 2(^2 + = 2[(a2 - + i(6, + < ) ] [> ( 7 )

£22 = Sn = 2(K ~ i k i ) = 2 [ K + * i ) + ~

tti = « i = £12 = i n = 2[*» ™

§4.5 SPECIAL TH E OR Y OF RELATIVITY 209

MaxwelPs eight equations read then in spinor language

. o r * = 2 v (5)

2. Fundamental Relativistic Invariants

1. Speed of light in empty space c

2. Length of a world line element ds :

ds2 = dx2 + dy2 + dz2 - c2dt2

3. Phase of an electromagnetic wave 4. Rest mass of a particle m 5. Electric charge e

6. Action integral j * * L dty if it is understood that t means the proper time

of the observer and that the Lagrangian L is meant to be,

L = mQc2(l — Vl—~W)-~ V (V is the potential energy),

7. Planck's constant h 8. Entropy S = S0 (S = entropy of moving system; S0 = entropy of

system at rest) 9. Boltzmann's constant k

10. Rest temperature T0 = T/Vl — £2

11. Heat Q0 = QlVr^ Bibliography

1. BERGMANN , P. G . , Introduction to the Theory of Relativity, Prentice-Hall, Inc., New York, 1942. Probably the best modern representation of the subject.

2. EDDINGTON , Sir A. S., The Mathematical Theory of Relativity, Cambridge University Press, London, 1923.

3. LAUE, M . V., Das Relativitatsprinzip, Vol. 1 (special theory), 4th ed., 1921; Vol. 2 (general theory), 2d ed., 1923; F. Vieweg und Sohn, Braunschweig.

4. LORENTZ , A. H., EINSTEIN, A. and MINKOWSKI , H., Das Relativitatsprinzip,

4th ed., B. G . Teubner, Leipzig, 1922. (An English translation, The Principle

of Relativity, is available from Dover Publications, Inc., New York 5. PAULI, W . , Jr., Relativitatstheorie, 4th ed., B. G . Teubner, Leipzig, 1922. (Article

in Enzyklopadie der mathematischen Wissenschaften, Vol. 5, Part 2, Sec. 4.) An exhaustive treatment of the earlier parts of the theory.

6. W E Y L , H., Raum, Zeit, Materie, 5th ed., Julius Springer, Berlin, 1923. Discusses the fundamental philosophical principles of space and time. (A translation was published by Dover Publications, New York, under the title Space, Time,

Matter.)

Chapter 7

T H E G E N E R A L T H E O R Y O F R E L A T I V I T Y

B Y H E N R Y Z A T Z K I S

Assistant Professor of Mathematics Newark College of Engineering

1. Mathematical Basis of General Relativity

1.1. Mathematical introduction. The world continuum of space-time is no longer considered " flat " or " Euclidean/' as was done in the special theory of relativity, but " curved " ' o r " Riemannian." This means there exists a " metric tensor " gQa — gaQ, such that the distance dr between two infinitesimally close world points (a;1,*2,*3,*4) and (x1 + dx1, x2 + dx2, xz + dxz, x* + dx4) is expressed by

dr2 = gQa dx<? dxa,

o = a (Y^- Y% ¥^ — t 6 pa — 6 cr@V ' > ' / ' — 1

whereas in flat space the distance is

dt2 = rjeadxe dxa (2)

The essential point is that the gQG are functions of the coordinates and that no special coordinate system exists in which gQa would be constant in a finite domain.

It was Einstein's fundamental idea to establish a relation between the metric tensor and the matter distribution in the world continuum. This relation is the content of Einstein's field equations. The gQa play the role of gravitational potentials and are the direct generalization of the Newtonian gravitational potential. More specifically, if the deviation from flatness is small, all gQO vanish, except gu, which then becomes identical with the Newtonian potential of a mass distribution.

Another important geometrical object in a Riemanian manifold, besides the metric tensor, is the Riemanian curvature tensor

r> a "PAR "PER "Titf T^ct) i P A "PTO / o \ ^XQT. — 1 T*,q ~ 1 TQ,X ~ 1 a>XL TQ T " 1 C O Q 1 TA W)

210

§ 1.2 THE GENERAL THEORY OF RELATIVITY 211

A comma denotes as usual the derivative with respect to the corresponding coordinate.

The r symbols are called the Christoffel three-index symbols of the second kind, and are defined by

The P s vanish identically if the gQa are constants, i.e., if the space is flat. However, even in flat space the gQa may not be constants as a result of a special choice of coordinates, e.g., if we choose polar coordinates instead of Cartesian coordinates in ordinary three-dimensional Euclidean space. It is therefore useful to have the following criterion for the curvature of a given space : The necessary and sufficient condition for a space to be flat is that all the components of the Riemannian curvature tensor vanish.

From the curvature tensor we can obtain tensors of lower rank by the process of " contraction." Of special interest is the tensor of rank two, denoted by RXa and the completely contracted curvature tensor, which gives us the curvature scalar R.

R=gX°Rxa ( 6 )

Written in terms of the Christoffel symbols, RXa takes the form

nXa — 1 crco,A — A aA,co ~~ 1 T O > A <TA " T 1 T A a <rA \ ' )

For later purposes we need the expression denoted by GKa

GXa = Rka — \g*"'R (8).

By contracting the above expression we obtain

gXaGx° = R-(±)-4R = -R (9)

Therefore GXa == 0 (10)

is equivalent to RXa = 0 (11)

Finally we shall denote the determinant of the gQa matrix by g.

1.2* The field equations. If matter can be represented by a continuous distribution p = p(xyy,z) which depends on space but not on time,

2 1 2 T H E GENERAL THEORY OF RELATIVITY §1.3

the gravitational potential G , generated by the mass distribution, is given by Poisson's equation

V 2 G = -4TT/> ( 1 )

In regions where the space is empty, this reduces to Laplace's equation

V 2 G = 0 ( 2 )

If the matter distribution depended on time also, it would be natural to assume that Eq. ( 1 ) should be generalized to the inhomogeneous wave equation

nG.= -4rrp(x,y9z,t) (3)

This equation, however, is not Lorentz-covariant, since the right side is not Lorentz-covariant. In order to be so, the right side should read p0 instead p, i.e., should include only the rest density. On the other hand, it is an empirical fact that the total mass is responsible for gravitational forces and not only the rest mass. Thus p must include the contributions from the kinetic energy as well. But the latter terms are no longer scalars.

Another possibility would have been to introduce four gravitational potentials and choose as the right-hand side the energy-momentum vector. But this is again not feasible. In the presence of matter, the mass density p is not a scalar, but one component of a tensor PXa (see the discussion of the stress energy tensor in the special theory of relativity).

In his search for the correct formulation Einstein postulated the field equations

GXa = - %>nKPXa ( 4 )

The GXa are the expressions defined by Eq. (8). K is the gravitational constant and PXa is the stress-energy tensor and represents a given distribution of matter. If the gravitating matter is concentrated in small regions, " mass points," and zero elsewhere, the field equations reduce to

GXa = 0 ( 5 )

which is equivalent to RXa — 0 ( 6 )

Equation ( 4 ) corresponds to Poisson's equation, and Eq. ( 5 ) to Laplace's equation. And just as Laplace's equation has solutions vv-hich are regular everywhere, except at the origin, also Eq. ( 4 ) possesses solutions gQa which are regular everywhere except for a singularity at the origin.

1.3. The variational pr inc ip le . Laplace's equation

V 2 G = 0 ( 1 )

§ 1.4 T H E GENERAL THEORY OF RELATIVITY 213

See EINSTEIN, A. and INFELD, L . , Canadian Journal of Mathematics, 1, No. 3 , 2 0 9 ( 1 9 4 9 ) .

is the Euler-Lagrange equation of the variational principle

8 J F ( | g r a d G | ) W = 0 (2)

As usual, G is varied only in the interior of the three-dimensional volume V and not on its boundary. Similarly, Einstein's field equations for empty space can be derived as the Euler-Lagrange equations of the variational principle

8 j D dx1 dx2 dx3 dx* = 0 (3)

Again the field variables gQa and their derivatives are to be varied only in the interior of the four-dimensional domain D.

If matter is present, the variational principle is to be generalized to

S j D (R + STTKM^11^ dx1 dx2 dx3 dx* = 0 (4)

where M is a function of the undifferentiated gQO and the distribution of matter. If PQA denotes the gravitational part of the stress-energy tensor, then the gravitational part of M9 denoted by M g r a v must obey the relation

a ( V - | M G R A V ) = P E F F ( ^ = V ^ K O ( 5 )

The electrodynamic part of M is given by

Mei = - g ^ < M e a ( 6 )

Here <J>ea is the antisymmetric tensor defined in the chapter on the special theory of relativity. In this case the field variables that are to be varied are the gQ(j and the electromagnetic potentials. Maxwell's equations are then part of the resulting Euler-Lagrange equations.

I A* T h e p o n d c r o m o t i v c law. Einstein, Infeld, and Hoffmann * have shown that the motion of bodies (considered as singularities in the field) follows from the field equations alone. This is in strong contrast to the Lorentz theory of electrodynamics where the ponderomotive law is an additional postulate, logically independent of the field equations of Maxwell. In the limiting case where the rest mass of the particle is so small that its contribution to the field is negligible, its motion under the influence of a gravitational field alone becomes indistinguishable from a force-free or

214 T H E GENERAL THEORY OF RELATIVITY §1.5

ds2

inertial motion. In curved space, it describes a geodesic line, which corresponds to a straight line in flat space. Therefore its path is

d*x* . VQ dx*dx°_

the equation of a geodesic in a Riemannian manifold. If the space is flat, all the T vanish and the classical law of inertia results :

-d^ = ° P)

1.5. The Schwarzschild solution. Only a few rigorous solutions of Einstein's field equations are known. One of them is the Schwarzschild solution, which describes the static field of a mass point at rest. Let m be its mass; then the equation of a line element in polar coordinates is

dr2 = c2(l - ^tyd* ~ ( l - d r % ~ r* de* ~ r* s i n 2 0 d<P (1)

We shall introduce dimensionless variables by letting

— = ds, —- = T, — = jR, 2oc = ^^tm = " gravitational radius " OL _ OC OC C*

and obtain

= (* ~ l l ) d T 2 l l ) 1 ~ R 2 dd* ~~ R 2 ^ 6 d<^ ^

H. Weyl and T . Levi-Civita * succeeded in finding the static solutions that have only rotational but not spherical symmetry.

1.6. The three "Einstein effects " +

a. The motion of the perihelion. Suppose that an infinitesimal particle is moving in the field of a Schwarzschild singularity at the origin. The equation of motion is

d2xe RQ dxx dxa m

The r can be computed from Eq. (45-2). One obtains a plane motion

* WEYL, H., Ann. Physik, 54, 117 (1917); 59, 185 (1919). BACH and WEYL, H., Math. Z.y 13, 142 (1921). LEVI-CIVITA, T . , Atti accad. naz. Lincei (several notes, 1918-1919).

t See WILSON, H. A., Modern Physics, 3d ed., Blackie & Son, Ltd., London, 1948, p. 392 (the treatment we follow here).

§ 1 . 6 T H E GENERAL THEORY OF RELATIVITY 215

which can be chosen in the plane 0 — 7r/2. T W O further integrals of the motion are

A = £[. = ^ (conservation of angular momentum) (2)

/ 2 \ dT

and A = ll — ~R) (conservation of energy) (3)

With the substitution, u = l/R, we get the equation of the orbit: *L + u = ±+V (4)

The corresponding Newtonian law is

The solution of Eq. (5) is

d 2 u . 1 t*\

u=fi2(l + ecos<f>) (6)

This is an ellipse if e < 1. The Einstein term 3w2 is very small compared to I / / / 2 in planetary motion. Therefore Eq. (4) can be solved by the methods of successive approximations. One substitutes on the right-hand side the Newtonian value of w, and obtains the differential equation

^ + " = ^ + i i ( i + 4 + 2 € C O S ^ + T 6 2 c o s 2 ^ ) The solution is

jj2+ Ji ( 1 ' + T ) ( 1 + € C O S ^ + & S I N ^ ~~ T 6 C O S 2 ^ )

( 7 )

(8)

If it were not for the term <j> sin <f>, u would have the period 2TT. The actual period is 27r + 8, where 8 is a small number. If this value is substituted in Eq. (8) and higher powers of 8 are neglected, one obtains

P _ 6TT _67ra ; 2 r 2 _ (mK2m2

~~ ^H2 ~ ^Wc^1 W

In the case of the planet Mercury, this formula gives for the rotation of the major axis of the orbit in 100 years approximately 43 seconds of arc. This is in good agreement with observation.

As A. Sommerfeld has pointed out, each deviation from the Newtonian or Coulomb potential causes a perihelion motion of the Kepler ellipse. In the theory of the fine structure of the hydrogen spectrum the deviation

216 T H E GENERAL THEORY OF RELATIVITY § 1 6

results from the mass-variability of the orbital electron. However the effect is much smaller than here. The rate of the precession of the perihelion is only about one-sixth of the rate produced here.

b. Deflection of light rays hy the sun. For a light ray path ds = . 0. Therefore h = °o. The differential equation for the path becomes

d2u W + » = W (10)

If the right-hand side were zero, the solution would be

w = cos </> (11)

where <f> is the angle between r and the x axis, the equation of a straight-line parallel to the y axis at a distance p. Again, we substitute this value for u into the right-hand side of Eq. (10). This gives

d2u _ 3 1 + cos 2 <f> !ftÛ~J2 2 (12)

The solution is

' = 7 c o s ^ + ^ ( t - - O - C O S 2 ^ ) ( 1 3 )

From Eq. (13) u — 0, or r — oo for <f> = ± (77/2 + £ ) , where £ is small. To find£, we have the equation

° — H ( M ) or S = — (15)

P

The asymptotic directions of the light ray differ by 2/p radians from the undeflected direction. The total deviation of a light ray, starting from infinity where the space is flat, and going off into the opposite direction to infinity, where the space is again flat, is

8 = 7 = T d = ( X p ) ( 1 6 )

c. The red shift of spectral lines. Consider two identical " atomic clocks," one on the surface of the sun and one on the surface of the earth, where the curvature of space may be assumed negligible. If both clocks are at rest, we have

dr==dd = dct>=0 (17)

§1.6 T H E GENERAL THEORY OF RELATIVITY 217

Hence dr2 = c2^l -*^dt* (18)

if the proper-time dt/c between two successive signals will be the same on the sun as on the earth. But the observed time intervals dt' (on the sun) and dt" (on the earth) will have the ratio

dt" _ L 2Km_ v Km dt' v c2r c2r

l _ ™ = ] / _ — (19)

the corresponding frequencies have the ratio

f 1 + ^ (20)

8 / = / " - / ' = fJ/' (21)

The frequency of the same spectrum line, emitted by an atom on the surface of the sun, appears shifted to the red as compared with the line emitted by the same atom on the surface of the earth.

Bibliography

See references at end of Chapter 6.

Chapter 8

H Y D R O D Y N A M I C S A N D A E R O D Y N A M I C S

B Y M A X M. M U N K

Department of Aeronautical Engineering Catholic University of America

1. Assumptions and Definitions

1.1. Theoretical fluid dynamics deals with continuous fluid masses. Modern physics interprets the continuous properties of such masses as statistical averages. The passage from the general equations governing the fluid motion to specific solutions often involves the replacement of the correct equations by special and simplified equations approaching merely the exact equations asymptotically as special additional conditions are more and more exactly complied with. These simpler and more special equations again lead to specific formulas. The student of hydrodynamics and of aerodynamics is particularly interested in these formulas. Pne should also recall that the velocity fields of hydrodynamics and of gas dynamics have much in common with the fields of electric theory and of the theory of elasticity, all being the solutions of the basic equations subject to boundary conditions. The student will therefore find comparison of the formulas of this section with those in other allied sections a useful and often profitable occupation.

1.2. We divide the fluid body into small compact portions called fluid particles, which operate under Newton's law of dynamics, notwithstanding the continual loss of identity because of perpetual diffusion.

1.3. We reconcile the exchange of momentum that arises from diffuse convection, § 1.2, by assuming that viscous stresses act within the fluid.

1.4. A pure pressure p, within the fluid, is the mechanism adapting the fluid density to the physical problem.

1.5. A perfect fluid is one free of diffusion and hence free of viscosity stresses.

218

§1,6 H Y D R O D Y N A M I C S A N D AERODYNAMICS 219

2. Hydrostatics

2 .1 . The buoyancy of an immersed body is equal to the weight of the displaced fluid.

2 .2 . The buoyancy force acts along a vertical line passing through the center of gravity (CG) of the displaced fluid.

2.3. The wholly submersed body is in stable equilibrium if it weighs as much as the displaced fluid, and if its center of gravity lies vertically below the center of gravity of the displaced fluid.

2.4. The metacenter of a floating partially submersed body is the point at which the body would have to be suspended to experience moments, when slightly tilted from equilibrium, equal to the hydrostatic moments.

2.5. The distance between the center of gravity of the body and the metacenter is called the metacentric height h.

2.6. Let a denote the vertical distance between the C G of the body and the C G of the displaced fluid, positive if the C G of the body is above the CG of said fluid. Let / denote the axial (not central) moment of inertia

I=frdA (1)

of the area A cut by the partially immersed body out of the fluid surface, the C G of said area having the ordinate y = 0. The metacentric height is then

M C H = / / F - a (2)

where V denotes the volume of the displaced fluid.

2.7. The pressure gradient of standing fluid is vertical and equal to the specific gravity of the fluid.

1.6. The fluid is called perfect in a stricter sense if its density is constant and unchangeable.

1.7. A perfect gas subordinates itself to § 1.5, it obeys the equation of state of a perfect gas.

plP = RB (1)

where 8= temperature, JR = gas constant, p = mass density, and its internal energy u per unit mass is proportional to the absolute temperature

u = kv6 (2)

220 H Y D R O D Y N A M I C S A N D AERODYNAMICS §3 . 1

3. Kinematics

3.1. The bulk or particle velocity v is a vector and a function of the time t and of the space vector r. The Cartesian components of r will be designated by x, y, z. The Cartesian components of v parallel to said space vector components respectively will be designated by u, v, w. This way of describing the flow is ordinarily designated as Eulerian.

3.2* The less common Lagrangian description identifies the fluid particles by suitable variables and follows the motion of each individual particle.

3.3. The divergence of a velocity field v is defined hy

V* v = dujdx + dvjdy + dw/dz (1) It is a scalar quantity.

3.4. The rotation or " curl " of a velocity field, also called its vorticity, is defined by the vector whose Cartesian components are

V X v = (uy - vxy vz — wyy wx —;uz) (1)

3.5. The gradient of a scalar field p such as the pressure is defined by

grad^ = Vp = (px,py,pz) (1)

3.6. The acceleration a of a fluid particle in terms of the Eulerian representation is equal to

a = .0v/0* + (v-V)v (1)

The first right-hand term is called the local portion of the acceleration, the second term is called the convection portion. The x component of the convective term is

uux + vuy + wuz (2)

and similarly the y and z components.

3.7. The convective portion of the acceleration can be transformed as follows.

( v v ) v = - | V ^ 2 + v x (VX v) (1)

3.8. A flow is called steady if its velocity and all its other properties are a function of the space coordinates only and not a function of the time t. The local portion of the acceleration dv/dt is then zero.

§ 3.9 H Y D R O D Y N A M I C S A N D AERODYNAMICS 221

The existence of a stream function is equivalent to the absence of divergence.

3.9. A-flow-field whose velocity v can be represented as the gradient of a scalar quantity cf> is called a potential flow, or a nonrotational flow. The quantity </> is called the potential of the velocity, or the velocity potential.

3.10. The rotation of a potential flow is zero.

V X V ^ O (1)

3.11. A velocity field v that has a potential cf> is given by the scalar line integral cf> = — \dr • v, taken along any line connecting a reference point with the point in question.

3.12. The scalar line integral — \dr • v taken along a closed line is called the circulation.

3.13. The scalar integral fdf* v, taken through a surface / , bounded or unbounded, is called the flux of v through / .

3.14. The flow field of an incompressible fluid has the divergence zero, V v = 0.

3.15. The general equation of continuity for any fluid is

V M = - 3 p / ^ (1)

3.16. The divergence of the vorticity of any velocity field v is identically zero.

V V X v= 0 (1)

3.17. The substantive rate of change of the circulation of v is that change along a closed line moving with the fluid particles, i.e., along one always occupied by the same fluid particles. This substantive rate of change equals the circulation of the acceleration

D/Dt \dr • v = \dr • a (1)

3.18. In a plane two-dimensional flow of an incompressible fluid, the flux through any line connecting a pair of points is independent of the shape of the line of connection. This flux is called stream function i/r if one of the two points is chosen as a fixed reference point and the other point is varied

u = difj/dy, v = — difjjdx (1)

222 H Y D R O D Y N A M I C S A N D AERODYNAMICS § 3.19

3.26. If the flow has a potential <£, then K is equal to

K=\ldvvp<t>

3.27. The momentum of the flow is equal to

M = $dS V (vp)r + $do • vpr

3.28. The moment of momentum of the flow is equal to

$dS vpr = \ \dSi?J X vp)r • r — \ jda x (pv)r • r

3.29. Biot Savart dSdS'CV* v ) x ( r - r ' )

3.19. Axially symmetric two-dimensional flows have no velocity components at right angles to the meridian plane. They have a stream function \jj if the fluid is incompressible

U = -~:K v=--yll>x (1)

where r denotes the distance from the axis of symmetry.

3.20. With a steady two-dimensional compressible flow the quantity v/>, called mass flux density, possesses a stream function ifj equivalent to §3.18 and § 3.19, respectively. If the flow is plane,

up = dty\dy, vp= - di/j/dx (1)

3.21. The plane having the velocity components u and v of a plane two-dimensional flow as Cartesian coordinates is called the hodograph plane.

3.22. The space vector xy y has a potential L in the hodograph plane « , v, provided the velocity vector u, v has a potential </> in the physical plane x, y, and conversely.

L =—</> + u - x + ©• y (1)

3.23. The stream function of (w, v) is the potential of the flow v> —u. The theorem of § 3.22 holds therefore also for the stream function.

3.24. In the following, S denotes space, dS=dxdydz; O denotes a closed surface, da an element thereof. A bounded surface is denoted by / .

3.25. The kinetic energy K of the flow v within the space S bounded by the surface O is equal to

§ 4.1 H Y D R O D Y N A M I C S A N D A E R O D Y N A M I C S

p = pressure dQ = heat differential p = mass density u = internal energy V ~ specific volume i = enthalpy, also designated by h 0 = temperature specific heat of constant volume s = entropy c p = specific heat of constant pressure

4.2. Notations, perfect gases :

R = gas constant y = isentropic exponent

4.3. du = —pdV + dQ == —pdV + 6 ds

4.4. dk= di= -V dp + 6 ds

4.5. Perfect gases s p = Rp6, u= Cv6y .h=CJ&

R ^ C v - C v = C P ^ ^ ^ C v ( y - l ) y

^ 71 C y Cy

where n denotes degrees of freedom.

s = CvInp — Cpln p

5. Forces and Stresses

5.1. There may act on the fluid external body forces / per unit volume. These forces may have a potential

/ = - v f

5.2. The buoyancy force per unit volume exerted on the fluid particle by a pressure distribution p is —V/>-

5.3. The force effect per unit volume caused within an incompressible fluid by the simplest type of viscosity is (/x denoting the modulus of viscosity)

(1)

4. Thermodynamics

4.1. Notations:

224 H Y D R O D Y N A M I C S A N D AERODYNAMICS §5.4

5.4. The viscosity stress within the incompressible fluid corresponding to the equation in § 5.3 is

s = /x[Vv + vV] = ®xx ®xy °xz ux vx wx "x Uy ®yx Vyz • Uy Vy Wy vx

Vy vz (1)

Gzx °zy <*zz uz vz wz Wy ™z\

5.5. The average normal stress crxx + & y v + c r z z of the stress §5.4 is zero.

5.6. The dissipation of kinetic energy into heat in accordance with § 5.4 is, per unit volume,

W « + /*(| V X v|)2 (1)

where a denotes the acceleration of the fluid.

5.7. The same dissipation for a finite portion of space S is

^Jdcj-fl + jLtJrfSdVX v|)2 (1)

5.8. If the velocity is, and remains, zero at the boundary of the space portion S, the total dissipation (but not the local dissipation, § 5.4) becomes equal to /x/2 times the integral of the vorticity squared over the space portion.

6. Dynamic Equations (External forces assumed absent)

6.1. Euler's equation for nonviscous fluids :

dv/a* + ( v V ) v ~ = - ^ - , ^-+VdV=0 P P

along any streamline.

6.2. Euler's equation for steady potential flows :

= - 4- V(« 2 + *>2 + «>2) P 2

6 .3. For a steady isentropic potential flow of an in viscid gas : , , u2 + v2 + w2

n0 — n -\

(i)

(i)

(i)

6.4. Stokes-Navier equation for a steady incompressible fluid having viscosity:

- l = ( r V ) „ _ i i v V f (1) p p


7. Equations of Continuity for Steady Potential Flow of Nonviscous Fluids (see also §§3.14, 3.15)

7.1. Laplace's equation for the potential of an incompressible fluid of constant density:

V ^ J ^ + S ^ f l (1)

1.2. For plane, two-dimensional flow:

^ + ^ = 0 (1) dx* + dy* K )

7.3. For polar coordinates, r, 6

*rr + i * M + ^ = 0 (1)

7.4. For spherical polar coordinates :

0 , . . . . 0 . 1 sin 0 - (r*<f>r) + m (sin Bfo) + -^jKa, = 0 (1)

x = r cos 0, y = r sin 9 cos co, z — r sin 6 sin a> (2)

6.5. Helmholtz' theorem : If external body forces and viscosity are absent, individual vortices move with the fluid; i.e., individual vortices always consist of the same fluid particles.

6.6. For plane two-dimensional flow of an incompressible fluid, each fluid particle preserves its vorticity.

6.7. In steady plane two-dimensional flow of a perfect gas along any one streamline, the vorticity is proportional to the pressure.

6.8. Convection and diffusion of vorticity: Plane two-dimensional flow

WT + v • = ' (1) py

6.9. In absence of viscosity and of external body forces, a homogeneous fluid executing a potential flow will continue to do so.

6.10. The continued existence of a potential implies that all dynamic equations and requirements are complied with.


7.10. In the hodograph plane:

( ' - ? H - + - ( ' - 2 S £ + S K + ( ' - ? ) * » - ° <•>

7.11. The stream function :

7.5. For a plane two-dimensional flow obeying Laplace's equation, the equation for the potential h of (x, y) as a function of (w, v) again obeys Laplace's equation.

x = dh\duy y = dLjdv

d2Lldu2 + 32Lldv2 = 0

7.6. The value of a solution <f> of Laplace's equation (§7.1) at any point is the arithmetical mean of its values over any sphere having that point as its center (in three-dimensional flows) or any equivalent circle (in two-dimensional flows).

7.7. Equation of continuity for a steady potential flow of a compressible fluid (see § 6.4).

^xx1 ~~^~) +cf)yy(i +^>zz(i ~^~) 2 2 2

where c denotes the local velocity of sound.

7.8. For polytropic expansion (y — expansion exponent):

2 (1 —(l>xi4>Xi)V ' V</> = ~l<l>xi<l>xjrf>xixk (!)

y — 1

(i,* = 1 , 2 , 3)

7.9. For two-dimensional flow in plane polar coordinates:

§7.12 H Y D R O D Y N A M I C S A N D AERODYNAMICS 227

7.12. Reciprocal potential, L:

~~ ~c^jûu ~~ ~tf^LVv + —JTûv = 0

8. Particular Solutions of Laplace's Equation

8.1. Sink or source in three dimensions:

1 477 r

8.2. Sink or source in two dimensions:

(1)

(1)

8.3. Rankine flow in three or two dimensions :

<T> = X+(F>11 X+cf>2 (1)

8.4. Flow past a sphere :

* = «o(r + $ ) c o 8 0 (1)

8.5. Flow past a circle:

'+=*o(r + y-) c o s d (!)

8.6. General solution suitable for solving problems relating to a straight line in two-dimensional flow :

6 = real part of \ (1) Y F i(z - V*2 - l)n

8.7. Complex variables: If u + iv is an analytic function of 'x + iy, then u and v are each solutions of Laplace's equation.

8.8. If f(x,y,z) is a solution of Laplace's equation, then

LF(±3L±\ (1) is also a solution.

8.9. Degree zero: The following is a solution, F and G denoting any analytic function.


1

« = — u (1) where u is given in § 8.9.

r

8.11. A conformal transformation of the system of lines </> = constant, ijj = constant in a plane leads again to a solution of Laplace's equation if the initial system represents potential lines and streamlines consistent with that equation.

8.12.^ The lines of equal potential and the lines of equal stream function form an orthogonal system.

8.13. Fundamental solutions of Laplace's equation in three dimensions: dn 1

^ = Jx^~r ^

The factor of rn expressed as a function of cos </> is called a Legendre function.

8.14. Kinetic energy of the potential flow of the incompressible fluid

T=-£-jdS » a = - £ , J d f a r . ^ (1)

8.15. Variational principle: The potential flow of the incompressible fluid has a smaller kinetic energy than any other thinkable potential flow of the incompressible fluid having the same values of the potential at the boundary of the region considered wherever there is a flux different than zero across the boundary.

8.16. Differentiation: The differential quotient d^jdx of a solution <f> of Laplace's equation with respect to a Cartesian coordinate x is again a solution.

9. Apparent Additional Mass

9.1. An immersed body, moving in a perfect incompressible fluid otherwise at rest, responds to external forces as if its mass were increased by an apparent additional mass. This mass depends on the shape of the body only and is, in general, different for motions in various directions relative to the body. (Translation.)

9.2. The apparent additional mass is equal to the kinetic energy of the fluid, divided by one-half the square of the velocity of advance.

8.10. Degree — 1 ; The following are solutions.


9.10. The additional apparent momentum is a symmetric and linear function of the velocity, i.e., at least three principal directions mutually at right angles to each other exist, for which the momentum is parallel to the velocity. The momentum in any other direction (relative to the body) can be computed from the velocity components in the principal directions by simple superposition.

9.11. The momentum transmitted from a body to a barotropically expanding gas surrounding the body, from the time both, the body and the gas, were at rest until the translational motion of the body as well as the motion of the gas has become steady, does not depend on the time history of the motion.

9 3 . The apparent additional mass is equal to the momentum transmitted from the body to the fluid from initial rest, divided by the velocity of advance.

9.4. The apparent mass K of a body having the same density as the fluid is equal to the static moment of the sink source system that would induce the fluid motion, multiplied by the density, or

K=p\dSrV'v (1)

9.5. The apparent additional mass is equal to the surface integral of the potential, multiplied by the density, or

^ = P W (i). 9.6. The apparent additional mass, divided by the mass of the displaced

fluid, is called the factor of the apparent mass.

9.7. Factor of apparent additional mass: Circular cylinder moving crosswise, 1; sphere,

9.8. Apparent additional mass: Elliptical cylinder of length L-><x>, moving crosswise at a right angle to major axis, a,

La^p (1)

Flat plate of width a, moving crosswise,

La^p (2)

Circular disk of diameter r, moving crosswise,


a) 10.2. Cross force per unit axial length (lift):

dL „ dS v*o . „ dS v*p — r — — 2 ~~ sin a cos oc= 2 — a dx dx 2 dx 2

10.3. The total air force of streamlined spindle is zero.

10.4. The total unstable moment of the air force (couple) is

v2p M = 2 volume a

10.5. The total lift of the spindle ending abruptly in a flat base of cross section S is

10.6. The pressure around any one cross section is proportional to the distance from the diameter normal to the plane of symmetry.

9.12. The energy for subsonic motion has a lower bound, associated with infinitely gradual setting up of the motion.

9.13. The minimum energy is composed of the kinetic energy and of the elastic energy of the gas.

9.14. We can evaluate the elastic energy if the total mass of the gas involved in the motion remains constant for all velocities of advance.

9.15. For polytropic expansion the elastic energy is the space integral of

[1 - M 02 ( y - l)p2/2]W<r-i> [1 - M0%y - l)p0»/2]W<y-i>

y ( y - i ) y

[1 - M 02 ( y - l)z;2/2]1/^-1>[l - Mp\y - l)v0*] ^ ( 1 )

r - i

where vQ = ambient velocity, v = local velocity, M0 = ambient Mach number, y = polytropic exponent.

10. Airship Theory

10.1. Assumptions: Perfect incompressible flow, slender airship body with circular section, no fins, small angle of attack, linearization, cross-sectional area, X-axial coordinate.


11. Wing Profile Contours; Two-dimensional Flow with Circulation

11.1. The plane two-dimensional flow past such a contour is determined by the ordinary boundary condition of zero flux in conjunction with Kutta's condition that the separation point coincide with the sharp-pointed trailing edge.

11.2. The resulting air force, or lift, acts at right angles to the translational motion of advance. The lift passes through a point of the contour called center of pressure and is proportional to the circulation about the contour.

L = vpT per unit span (1)

11.3. The profile possesses a point called areodynamic center. The moment of the air force with respect to this center is not dependent on the angle of attack but is merely proportional to the density and to the square of the velocity of advance.

11.4. The aerodynamic center of a thin contour extending close to a straight line, called the chord, is positioned near the 25 % point of the chord; that is, it is spaced one-quarter chord length downstream from the leading edge.

11.5. The direction of forward motion for zero lift of a slender profile is nearly parallel to the line connecting the trailing edge with the center of the profile, this center being at the 50 % station and equidistant from the upper and lower camber line, or contour boundary.

11.6. Lift coefficient:

L cvlp\2

where L == lift per unit span, v = velocity of advance, c p = density, oc = angle of attack; dcL\doc is called the lift slope.

11.7. As the angle of attack increases, the CP (center of pressure) approaches the aerodynamic center (AC) with positively cambered profiles.

11.8. If the CP does not travel but is fixed, it coincides with the A C

11.9. The linearized theory applies to slender airfoil sections. The boundary condition of zero flux at the contour is replaced by prescribing the flow direction along the chord, namely, to be parallel to the average

(i)

= chord,


camber direction at the chord station. Only terms linear in the perturbation velocity components are retained.

11.10. The angle of attack of zero lift is given by the integral

J - i ( l -x)Vl - *2

where chord = 2, x = chord station, £ = mean camber.

11.11. The angle of attack for zero moment with respect to the 5 0 % station is given by the integral

2 r+1 x£ dx oc= — - •> (1)

11.12. In actual, slightly viscous air the profiles experience, furthermore, a resistance or drag, called the profile drag.

12. Airfoils in Three Dimensions

12.1. Assumptions: The spanwise extension of the airfoil or airfoils is large when compared with the flowwise extension. The ratio of the square of the span, or maximum span, to the total projected airfoil area is called aspect ratio. Viscosity is absent. The air, having been in contact with the upper airfoil surface, slides sideways over the air having been in contact with the lower airfoil surface. The vortices representing this condition coincide with the streamlines and are assumed to extend straight downstream. There occurs a drag called induced drag.

12.2. The spanwise lift distribution for minimum induced drag requires constant downwash velocity along the span or spans, provided all chordwise distances are reduced to zero; that is, the airfoil or airfoils are replaced by the flowwise projection.

12.3. The minimum induced drag is equal to

where k denotes the area of apparent additional mass of the front projection of the airfoils.


12.4. The minimum induced drag of a single airfoil, straight in front view, is

L 2

where b denotes the span. The lift slope of the single airfoil is equal to dCL_ 2TT . doc ~~ 1 + 2S/b*

12.5. For the single airfoil the relation of the center-of-pressure travel to the variation of the lift coefficient is the same as with the two-dimensional flow (infinite span).

13. Theory of a Uniformly Loaded Propeller Disk (Incompressible, nonviscous, infinite number of blades)

13.1. The theoretical efficiency, i.e., the upper bound of the uniformly loaded disk of a propeller is

17 = — = (1) 1 + Vl + 8 7 > 2 Z ) 2 7 7 p

where T = thrust, D = diameter, v = axial forward velocity.

13.2. The minimum power requirement for a hovering propeller (v = 0) is equal to

P=V2T*IPD27T (1)

13.3. The local slipstream velocity ratio in the region of a propeller is equal to

» . = y V[l + ( 8 T / ^ D « 7 r ) . - 1]

14. Free Surfaces (Incompressible, nonviscous, plane, two-dimensional)

14.1. The intrinsic equation for flow from all sides into an open channel of constant width 4 is

s = — — In c o s — (1) 77 Z

y = ±-6-sm&) (2) 77

where s = length of curve, 0 = direction of curve bounding stream, coefficient of contraction = j .


77 = 0.611 (2)

7 T + 2

14.3. For an infinite stream impinging on a strip of width one, with a cavity pressure equal to undisturbed pressure,

1 1 M

J ~ 1 + 7 T C O S 2 0 ( )

- ^ = ^ = 0 . 4 4 0 (2) pV 2 77 + 4 W

15. Vortex Motion

15.1. The condition for steady vortex motion in the absence of external forces is

V X [v X (V X v)] = 0 (1)

15.2. Bernoulli's equation holds for steady vortex flow, provided all vortices are parallel to the streamlines.

15.3. The force on a cylinder in laminar straight motion of constant vorticity is

/=2KPU (1)

where #c = circulation of circumference.

15.4. The stagger ratio of stable pairs of vortices (Karman's vortices) is

KTT = b/a, cosh2 KIT = 2, KTT = 0.8814 (1)

where a = distance of vortices in each line, b = distance between the two lines.

15.5. The velocity of translational advance of a circular vortex depends on the local distribution of the vorticity within the region of the ring.

14.2. For a slot in a wall of width 2 + 4/77,

5 = — A In ( - c o s 0) . (1)

where final width of the jet is 2. The coefficient of contraction is

§ 16.1 H Y D R O D Y N A M I C S A N D AERODYNAMICS 2 3 5

16.5. Tidal waves, shallow depth h:

c=Vgh (1)

16.6. Water jumps associated with tidal waves, steady :

^ 2 = - y (î2 + ^22) (1)

16.7. Same, relation between heights:

h2= hx 0)

where M denotes equivalent Mach number, v/c.

16.8. Progressing wave traveling in one direction only: The kinetic energy is equal to the potential energy.

17. Model Rules

17.1. Froude's rule:

Lgfe2 — constant (1)

where L = length, v = velocity, g == acceleration of gravity.

16. Waves (One-dimensional, weak)

16.1. Group velocity cg :

c° = c - x d \ ( 1 )

where c = wave velocity, A = wavelength, g = acceleration of gravity.

16.2. Surface waves, gravity, infinite depth:

c= VÂ/2T7 (1) 16.3. Capillary waves:

c = VlirkjpX (1) where k = surface tension (force/length).

16.4. Gravity and capillarity in unison:


17 .2. Reynolds' rule R ,= Reynolds' number = = — r - = constant (1)

V flip

where p — density, /x = modulus of viscosity, v = pu/p kinematic modulus of viscosity.

17.3. Mach's rule:

Mach number == M = vjc = constant (1)

where v = velocity, c = velocity of sound (see § 23.2),

17.4. Grashoff's rule:

pL*gMlv26 ==. const. (1)

where 9 = temperature, /3 — thermal expansion.

18. Viscosity (Incompressible)

18.1. Absence of inertia forces:

/XV 2 ^=V/), V - © = 0 (1)

18 .2. Sphere with radius a advancing with velocity U:

P=6rrii*U (1) where P = resistance.

18.3. Circular disk having radius r moving broadside on

R = 16r/977 (1)

R denotes the radius of the sphere having the same resistance.

18.4. Circular cylinder, two-dimensional flow:

l - y - l n ^ a / ^ ) U

where y = resistance per unit length.

18.5. See §§ 5.6 to 5.8; also Section 24.

18.6. Poiseuille flow through straight vertical circular pipe (laminar):

where a = radius of pipe.


19. Gas Flow. One- and Two-Dimensional (Isentropic)

19.1. Velocity of sound: c*= dp/dp (1)

If p is not a function of p, both must be varied for the particle considered. With steady flows they have to be varied along the streamline.

19.2. Velocity of sound for a perfect gas :

c* = yPtP = ygm (1)

19.3. General equation for sound wave motion :

dtyldT2 = c2V2cf> (l)

19.4. The kinetic energy of a simple, harmonic sound wave is equal to its elastic energy.

19.5. Force necessary to vibrate a sphere (linearized, amplitude << radius) :

\2 + k2a2 du . / 4 3 -ou j 4 + kW dT~r4 + #o*

where a = radius of cylinder, k = 27r/wavelength, a = 27r/period.

19.6. Subsonic flow past a circular cylinder (perfect gas):

- £ = OO + M ¥ I O + - » = (r + y ) c o s 0

(1)

+ M 2

+ (-4"r~i+ r""3)cos34+

(1)

(2)

18.7. The velocity profile across the diameter of a Poiseuille flow is a parabola.

18.8. Couette flow between two parallel plane walls sliding relatively to each other, steady flow:

Tangential force per unit area = a — pjv\a (1)

where a = distance of walls, v = difference of wall velocities.


y — 1 y — 1

2 y - 1

c 0 == escape velocity (2)

Equation 19.10 holds throughout centered flow, that is, waves traveling in one direction only.

19.11. Coparallelism: All Mach line systems of plane two-dimensional potential flows of the same polytropic gas are coparallel. That is to say, corresponding sides of the diamonds formed by corresponding pairs of adjacent Mach lines are parallel. Mach lines are corresponding if they are parallel at the same Mach number station. If they are parallel at one Mach number station, they are at all Mach number stations.

19.7. Characteristic equations for plane two-dimensional potential flows :

de±dp^pt = o (i)

For plane two-dimensional flow :

dvjv = ± tan p, d6, sin /x = 1/Af (2)

19.8. Same for polytropic expansion, integrated along the Mach line:

/* + V y ^ T t a n _ 1 V y - ^ l ^ ^ i ^ 0 W

For Prandtl-Meyer flow the characteristic equations (19.7) or (19.8) hold through the entire flow and not merely along any characteristic.

19.9. Characteristic equation for axially symmetric flow:

j n , , sin zx cos a _^ dr sin 6 sin /x _ dd ± dp P

2 R =F .—7a | p v = 0 (1)

pc* r sm (a i /x) v 7

19.10. Characteristic equation for unsteady one-dimensional flow:

2 2 - c ± = constant = ^ c 0 (1)


20. Gas Flows* Three Dimensional

20.1. Isentropic supersonic flow past a circular cone, symmetric (Taylor-Maccoll) :

,*> = ««,' (1)

where to = polar angle, u = radial velocity, v = peripheral velocity.

20.2. Velocity component of general linearized conical flow, M 2 = 2 :

U=F(—* + IY -)+F(—xriy) ) a) \iz + V * 2 + J 2 - W \iz + Vx2 + y2 — zV

(See §8.9) . 20.3. Throat flow:

where £ = radius of curvature of sonic line, r = throat radius, R = radius of curvature of nozzle wall, y = expansion exponent.

20.4. Reversal theorem for three-dimensional airfoil flow (linearized equations). The lift slope of any airfoil remains unchanged if the velocity of advance is reversed.

20.5. Slender delta wing, having span b. The lift coefficient slope computed with b27r/4 and v2p)2 as reference quantities is equal to 2.

20.6. Initial lift slope of slender, flat-based, finless missile body of circular cross section (linearized equations). The lift coefficient slope computed with D 2 7T /4 and ^2p/2 as reference quantities is equal to 2. (D = base diameter.)

20.7. Canonical variables: Every steady flow of a polytropic gas can be expressed by four dependent variables u\ay v/a, wja> and p, where a denotes the ultimate velocity of the streamline in question. That is to say, every streamline pattern serves an infinity of flows, in that the density can be prescribed arbitrarily at one point at each streamline. The flow may include shock waves.

21. Hypothetical Gases

21.1. Chaplygin gas:

p = - ± + B, y = - l ( 1 )

This gas can execute a plane, simple, sound wave of finite magnitude with


unchangeable contour. In plane two-dimensional potential flows all Mach line elements along any one Mach line of the conjugate family are parallel. The characteristics in the hodograph are straight. In the hodograph, the equation for the stream function is reducible to Laplace's equation or to the simple wave equation.

21.2* Munk's gas:

All Mach lines in the plane two-dimensional potential flow are straight. In the hodograph the characteristics are circles passing through the center. In the hodograph, the equation for the Legendre reciprocal potential is reducible to Laplace's equation or to the simple wave equation. This is not a polytropic expansion.

2 1 3 . Isothermal expansion:

y = + i (i)

Many equations for polytropic gases do not hold for this special case, but must be replaced by special equations containing logarithmic or exponential expressions. The ultimate velocity for this expansion is infinite, and the velocity of sound is constant. This constant velocity of sound is preferably used as reference velocity. The elastic energy of expansion available is unlimited.

21.4. Any expansion law p = f(p) may be replaced by

P = f(p) + constant (1)

without any change of the dynamics of the gas.

22 . Shockwaves

22 .1 . Steady, normal shock: a. Rankine-Hugoniot relations :

u12 + h1=u2* + h2 (energy) (1)

W j p ! = u2p2 (continuity) (2)

Pi + Piui =p2 + P2U22 (momentum) (3) where subscript 1 denotes upstream shockwave; subscript 2 denotes downstream shockwave; a denotes ultimate velocity.


oi — ^02> M i w 2 — c * 2

M, ( y - l W + 2 ^ M x ' + 5

2 y M 12 - ( y - l ) 7 M X

2 - 1 v r ; w

ft 2 y M ^ - ( y - l ) _ 7 M , ' - 1 ft- y + 1 _ 6 W-i-V W

P2P1_ PlPi

(7)

(8)

P8 (y + 1) (PM + (y - 1 ) = (6fa/>i)+1 / v = s ] 4 x m

pi ( y - i ) ( P 2 / / > i ) + ( y + i ) (/>2//>i) + 6 ^ l ' v ( y )

M = h W - f r - i ) . y h H i W ( 1 0 )

c , y + l X ( y - l ) M 12 + 2 V ;

where = i?/y — 1. 22.2 . Oblique shock waves : In equations (22.16-5), (22.1b-6), and

(22.1b-7) substitute M sin a for M. Then ~ 2M,2(sin a cos a — cot a) ...

t a n 8 = 2 + M 12 ( y + l - 2 s i n 2 a ) ( 1 )

cos a = v2 cos (c — 8) (2)

ôblique = ^normal + COS2 a (3)

sin a • fl2 sin (or — 8) = ^ | f o r m a l (4)

where cr = shock angle, normal shock a = 90°, 8 = deflection angle.

23. Cooling

23 .1 . Notation: k = conductivity, c 3 ) = h e a t capacity (specific heat), JUT = modulus of viscosity, L = reference length.

b. Polytropic expansion, perfect gas:

c ~ a i r + i 2 y..+ i

If M1 is given,


23.2 . Prandtl number: a = fic/k (1)

23.3. Nusselt number: gLi[k(et-e$ (l)

23.4. Impact temperature increase (for air) is about

A0 = (MPH/lOOf centigrade (1)

23.5 . Impact temperature increase

A0 = z>2/2^, cv ~ 1000 m2/sec2 (air) (1)

24. Boundary layers

24 .1 . Notations: u= streamwise velocity component, U = undisturbed velocity, x = streamwise coordinate, X= streamwise distance from leading edge, p == density, p, = modulus of viscosity, v = /x/p, R = Reynolds' number = XU[v, T= local shear, S = boundary layer thickness, S 9 9 same up to where velocity equals 99 % of undisturbed velocity, hd = asymptotic displacement of streamlines, S = " wetted " area.

24.2 . Boundary layer equations (incompressible):

uT + uux + vuy = — + tt„YV (1)

wx + ^ = 0 (2)

a. Boundary layer of flat plate laminar). Boundary layer is laminar, for R = 350,000 to 500,000.

T= 0332u2pVpLl(UPX) = 1.66>*/89 9 (1)

S99 = wc/i* (2)

§dispi= l.lWvxju (3) Total friction

Df = 132&Spu2l(2Vxu/^) (4)

b. Boundary layer of flat plate turbulent). The following holds for R= 1,000,000 to 10,000,000.

0.37* v ux\v


T = 0.0228pw2 (2)

= 0.i (3)

D, = 0.072p«2 y (4)

Bibliography 1. BATEMAN, H . , Partial Differential Equations, Cambridge University Press, London,

2. COURANT, R. and FRIEDRICHS, R. O., Supersonic Flow and Shock Waves, Inter-science Publishers, Inc., New York, 1948. Emphasizes the mathematical aspect of gas dynamics.

3. DURAND, W . F. , Aerodynamic Theory, Julius Springer, Berlin, 1935. See especially Vols. 1, 2, 6. (Dover reprint)

4. GOLDSTEIN, S., Modern Developments in Fluid Dynamics, Clarendon Press, Oxford, 1938.

5. LAMB, H., Hydrodynamics, 6th ed., Cambridge University Press, London, 1932. (Also Dover Publications, New York, 1945.) The classical treatise, rich and profound.

6. SAUER, R., Einfiihrung in die theoretische Gasdynamik, Julius Springer, Berlin, 1943. 7. TIETJENS, O. G., Applied Hydro- and Aeromechanics (based on lectures of L. Prandtl;

trans. J. P. Den Hartog), McGraw-Hill Book Company, Inc., New York, 1934. Short and clear. (Dover reprint)

8. TIETJENS, O. G., Fundamentals of Hydro- and Aeromechanics (based on lectures of L. Prandtl; trans. L. Rosenhead), McGraw-Hill Book Company, Inc., New York, 1934. Short and clear. (Dover reprint)

1944.

Chapter 9

BOUNDARY VALUE PROBLEMS IN MATHEMATICAL PHYSICS

B Y H E N R Y Z A T Z K I S

Assistant Professor of Mathematics Newark College of Engineering

1. The Significance of the Boundary

1.1. Introductory remarks. Boundary value problems occupy a central position in mathematical physics. T o give a systematic development of this subject would be well beyond the scope of the book, and we shall therefore confine ourselves to the discussion of some of the equations that occur more frequently. Other specifications will appear in other chapters. The major difficulty is not so much to find a solution of the given differential equation (in fact the equations admit always an infinity of solutions), but to find that solution which fits the given boundary values. If the solution depends on space and time, the latter can be treated formally as an additional coordinate in a space of one extra dimension, and the initial conditions thus also form part of the " boundary " conditions.

We are mainly concerned here with the Laplace equation, wave equation, and heat conduction equation. They are all linear, homogeneous, partial differential equations of second order. We plan first to discuss these equations individually and then to describe some of the features common to all which furnish unifying principles for their solutions. Finally, we shall also add some remarks concerning the corresponding inhomogeneous equations. A powerful tool to solve many important linear, homogeneous, differential equations is the method of separation of variables. However, it is not the only possibility, as will be seen subsequently.

1.2. The Laplace equation. The three-dimensional Laplace equation in Cartesian coordinates is defined by

244

§ 1 . 2 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 245

If <J> is independent of z, we obtain the two-dimensional Laplace equation

A function <D satisfying the Laplace equation is called harmonic. An intuitive interpretation of V 2 ® is the following. Consider a point P 0 at which <& has the value O 0 . Draw a little sphere around P 0 with a radius sufficiently small that we can use the Taylor formula to express the values of <D on the surface of the sphere and neglect third-order and higher-order terms. If we compute the arithmetic mean of <D on the spherical surface and denote it by <D, then <D — ® 0 = (V 2 O) 0 . We can therefore say that V 2 ® is a measure for the deviation of the value of ® at a given point from its average or " equilibrium " value. This interpretation is especially helpful in understanding the nature of the wave equation or heat conduction equation.

The Laplace equation is fundamental for potential theory; ® then represents the potential produced by gravitational masses (or electrostatic charges) in a region free from mass (or charge). The simplest boundary value

FIGURE 1

problem is the following: Given a simply connected domain D (Fig. 1), bounded by a continuous boundary B, the values of ® on B are prescribed to be a continuous function ®. We then wish to find the values of ® in the interior of D, satisfying the Laplace equation and taking on the prescribed boundary values. This is the " i n n e r " Dirichlet problem. In place of the function values ® themselves, their normal derivatives d<$>jdn may be prescribed on B. This constitutes the Neumann problem or, finally, a linear combination of the function values and their normal derivatives with constant coefficients may be prescribed on B. This is the " third " problem, which occurs mainly in connection with the heat conduction equation. The

246 BOUNDARY VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS § 1 . 3

condition, imposed on the solution O to be harmonic in Dy has, as a consequence that O and dQ/dn cannot be prescribed independent of each other and, therefore, cannot be prescribed simultaneously in an arbitrary manner.

If we wanted the solution O outside the domain Dy with the same boundary conditions, we should speak of the corresponding " outer " problems.

We understand that the positive direction of the normal always extends into the interior of the region D. T o give meaning to the concept of " normal, " we must assume that the boundary consists of at least piecewise continuously differentiable sections.

We finally remark that the domain may also be multiply connected.

1 3 . Method o f separation o f variables. Frequently the boundary is of a simple geometrical structure, e.g., a rectangle or a circle. Then we find it useful to employ rectangular or, more generally, curvilinear coordinates in which the boundary consists of sections along which one of the coordinates has a constant value. In case of the rectangle we use of course, the Cartesian coordinates themselves. The boundary consists then of segments, described by equations x = constant and y = constant. In case of a circular boundary of radius a, we use polar coordinates, and the boundary is given by r = a. The Laplace equation, after transformation to the new coordinates uy vy wy

usually yields to a solution of the form

®(uyv,zu) = U(u)V(v)Ww) (1)

We obtain then a system of ordinary differential equations for the functions Uy V, and W. The final step consists of obtaining any appropriate linear combination of the solutions of type Eq. (1) with constant coefficients chosen to satisfy the given boundary conditions. Some examples are the following.

a. Cartesian coordinates, three-dimensional: 32<D a2<£ d2<D '

Particular solutions: t>k,l>m = (kx + ly + mz), ( A » + P + m » = 1 ) . ( 3 )

b. Cartesian coordinates, two-dimensional:

32<D , a2<D A

Particular solutions: ^ = e±kix±iv) (5)

§ 1.3 BOUNDARY VALUE PROBLEMS I N MATHEMATICAL PHYSICS 247

! + a2) (w2 + a2) (w8 + a 2

3; = (12)

( A a - f l a ) " ' ( ^ - f l a )

(«! + ^ ) (tf, + b2) K + &*) ( < * - A 2 ) ( u3> b2 > u% > c 2 > %. The surfaces % = constant, # 2 = constant, w3 = constant represent confocal ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets, respectively.

c. Spherical coordinates (r>6,<p):

l a / 2 a o \ l a / . 0 l a2®

r 2 dr \ dr. j + r2 sin 0 a<9 \ 8 m dd J + r 2 sin2 0 " 0r* " " 0 ( 6 )

Particular solutions:

®m,,= (Arm + ^y±^Pm*cos6) (7)

where P mf c ( cos 0) = associated Legendre polynomial.

d. Cylindrical coordinates, three-dimensional (p,cp,z):

' • "aj? " + p ap + p2 a 92 + a*2 ~ ( 8 )

Particular solutions:

<D*,«i = e^^]m(ikp) (9)

where J m = Bessel function of order m.

e. Cylindrical coordinates, two-dimensional (p,<p): 820 i a<D l 8*$ ^ " + 7 " ^ + 7 " V - ° ( 1 0 )

Particular solutions : <$k= p±keik<? (11)

f. Elliptical coordinates (uvu2,uB):

The relations between the Cartesian and elliptical coordinates are :

248 BOUNDARY VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS § 1 . 3

Then ux = 0 in the ellipsoid

°?L i ?L j _ = i a2 ^ b2 ^ c2

If R< = V(Ui + a 2 ) (ii, + b2) (Ui +• c 2 ) , (f = 1,2,3)

the Laplacian reads as follows.

V 2 3 > = T ^ — — ^ r \(u2 - Us)RY

(«! — u2) (u2 — w3) (1*3 — ttj) [ a«! \ ;

> (13)

+ K _ « 2 ) i ? 3 _ (**JS) = 0

As an illustration, let us compute the potential of a charged ellipsoidal conductor, whose surface is given by

a2 ^ b2 ^ c2

The conductor must be a surface of constant potential. For large distances the potential must fall off as Q/r, where Q is the charge of the conductor. The equipotential surfaces are confocal ellipsoids, i.e., <D depends only on ux. The Laplace equation reduces to

The solution is

where R^g) = V(F+ a2) ( f + b2) ( f + c2). For large values of r*= x2 +y2 -\- z2, the value of £ approaches r 2 , and the potential O itself approaches the value

* ± y (16)

Therefore 2k = Q = charge of the conductor, and the, solution becomes

§1 .4 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 249

1.4. Method of integral equations. We can express the interior Dirichlet problem as the solution of an integral equation. Let Q and T be points on the boundary B and P, and interior point of the domain D; let n be the interior normal of P, andg(T') be the preassigned boundary values of ®(P) . The solution of the problem is then in the three-dimensional case

*m=JL,/(e)4-(4-) (i)

where f(Q) is the solution of the integral equation 3 2 . / ( T ) + / / B / ( Q ) A ( _ L ) , 5 ^ , ( r ) ( 2 )

In the two-dimensional case the corresponding formulas are

Uiog-L iQ \ rTQJ

dS. Q (3)

and

W) + jBf(Q)£-Q(^ySo = S(T) (4)

In the second case B is the curve bounding the area D, dSQ is a line element, and the line integral has to be taken in the positive sense (the interior of the domain must be to the left of B).

The integral equations for / are Fredholm integral equations of the second kind. For the methods of their solution the reader is referre*d to any textbook on integral equations or potential theory.

In the special case where the boundary is a sphere or circle of radius 1, an explicit solution can be given. The formulas are known as the Poisson integral formulas:

and

w [ 4 (log 4 ) - 2 7

(5)

(6)

1.5. Method of Green's function. This method is closely related to the foregoing discussion. The Green function G belonging to the domain D with the boundary B is defined in the following way: G is a function of two points P(#,YYZ) and Q(£,RJX), of which the point P varies in the interior of D, and Q varies in D + B. Also, G has the property that, as a function of Q, it is harmonic in D + B> with the exception of P. In P, the function G becomes singular, like l /r P Q.

250 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS § 1 . 5

Therefore, G can be written

G(rpQ) = ^ - + ^(rPQ) (1)

where TPQ is harmonic in the closed domain D + B. Finally, we assume that, on the boundary, co takes on the value — 1/?>Q. In the two-dimensional case, the procedure is identical, except that we replace 1/*>Q by log (1/*>Q).

The solution of the interior Dirichlet problem then becomes

in the three-dimensional case, and

in the two-dimensional case. As in the preceding section, the function g(Q) denotes the prescribed boundary values of <P.

For a sphere and a circle, Green's function can be stated explicitly. Let Q (Fig. 2) be a point on the boundary, and P an interior point whose

FIGURE 2

distance from the center O is R. Let P' be the " conjugate " point of P

with the distance R' from O, i.e., OP ~OP' = RR' = / 2 (/ = radius). Then

G(PQ)=\!r-l/Rr'

for the three-dimensional case, and

G ( P 0 = l o g ( l / r ) - l o g ( P r ' )

for the two-dimensional case. W. Thomson's (Lord Kelvin's) method of electrical images is based on these Green functions.

§1 .6 BOUNDARY VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 251

1*6+ Additional remarks about the two-dimensional case. The two-dimensional Laplace equation has an important property that any analytic function

f(z)=®(x,y)+iY(x,y) (1) is a solution of the Laplace equation. The fact that f(z) is analytic permits us to use all the results of function theory. Especially important is the theorem that an analytic function remains analytic under a conformal mapping. The conjugate functions O and W satisfy the Cauchy-Riemann equations

d®ldx=dx¥jdy, 3®ldy= -cWldx (2)

A consequence of Eq. (2) is that both <P a n d T satisfy Laplace's equation. The two families of curves O = constant and T = constant are mutually orthogonal. We can, therefore, interpret them, respectively, as lines of constant potential and lines of force. Under a conformal mapping they will transform into two families of mutually orthogonal curves. We utilize these results in solving problems in electrostatics and in the steady, two-dimensional flow of an incompressible, ideal liquid. If this flow is also irro-tational, the velocity distribution can be described by a velocity potential function <$>(x,y) whose Laplacian is zero. The Cartesian components of the velocity at the point (#,3/) are given by

u = d®ldx, v = dO/dy (3)

As an example let us consider a uniform motion with a constant velocity a in a positive x direction. Then

O = ax (4)

and, therefore, W = <xy (5)

or f(z) = oc(x + iy) = ocz, (z = x + iy) * (6)

The mapping a> = z2 maps the upper half-plane into the upper right-hand quadrant. As we can see from Eq. (8), the new streamlines are hyperbolas with the coordinate axes as asymptotes. The velocity potential takes the form

£ = oc2x2-y2) (9) The velocity components are

u = d^jdx = 2OL2X, v = d^jdy = 2oc2y (10)

* The mapping co = z2 gives us again a possible flow. If we write co = £ + ( 7 )

the new streamlines are given by t\ = loCxy = constant (8)

252 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS §1.6

The magnitude of the velocity is

Vu2 + v2=2oc2r, (r2 = x2+y2) (11)

In the plane every analytic function represents a conformal mapping. In three-dimensional space the possibilities of conformal mapping (i.e., angle-preserving mapping) are much more restricted. Aside from the similarity transformations (i.e., rigid rotations, translations, and stretching the x, y9

and z axes by the same factor), the only nontrivial conformal mapping of the space upon itself is the inversion at a sphere. If the radius of the sphere is equal to 1, the image point P ' of the object point P lies along the infinite half line through the center of the sphere and P, and its distance from the center is given by OR • OR' = I2. If the center of the sphere is at the origin, the coordinates of the point P' and P are, therefore,

I2 , I2 , I2

I (12)

X = P 2 " * ' y = -j?y> z =~R2~Z

i2 , i2 i2

As W. Thomson discovered, we have here also a possibility of obtaining new solutions of the Laplace equation from the original one by conformal mapping. However, we find one difference that distinguishes the three-dimensional from the two-dimensional case. If

,2 . ^ = 0 (13) 32F d2F d2F 'dx2 + 'dy2 + A ?

then F(x\y\z') does not satisfy the Laplace equation, but

does:

Ff^^F(x\y\zf) (14)

(dx')2 ^ (a>02 ( ^ ' ) 2

Green's function is closely related to the problem of conformal mapping in the two-dimensional case. Let £ = f(x + iy) be an analytic function that maps conformally the domain D in the z plane (z = x + iy) in such a manner into the unit circle in the y plane, such that the point £(£,17) in D goes over into the center of the unit circle; then — (1/2TT) log \f(x + iy) \ is the desired Green function. Thus we know the Green functions for all those domains which can be mapped conformally into the unit circle.

§1-7 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 253

1.7. The one-dimensional wave equation. The fundamental equation for the propagation of a wave in an anisotropic and homogeneous medium is

1 d2® ™ ' y t ] r (1)

where c is the velocity at which a disturbance will travel. If the direction of propagation is in the x direction only, the equation above reduces to

32<D 1 d2<£ dx* c* dt2

(2)

Just as was the case for the two-dimensional Laplace equation, we can write the general solution at once:

<D(x,J) = F(x - ct) + G(x + ct) (3)

where F represents a disturbance traveling in the positive x direction at velocity c, and G is a disturbance traveling in the negative x direction at the velocity c.

We want to apply this solution to find the motion of a vibrating string which is held fixed at its boundary points x = 0 and x = L, and whose initial displacement is prescribed to be <D(#,0) =• f(x)> and its initial velocity distribution

From the initial conditions it follows that

f(X) = Fix) + G(X) | ^ g(x)=-cF'(x) + cG'(x) ) - - K>

where primes denote derivatives with respect to the argument. If the second equation is integrated and combined with the first we obtain

*X*) = y (ft*)" T ilj^du) for O^x^L (5)

G(x) = ^ f ( x ) + ^J^g(u)du)j

The lower limit x0 is arbitrary, and it will be cancelled out if Eq. (5) is substituted into Eq. (3 ) :

O(*,0 = 1 1 cXJrCi

f(x - ct) +f(x + a) + — \x_ctg(tt)du (6)


Since / and g are defined only for values between 0 and L, the formula above is valid only for

ct ^ x t^L — ct

However, the boundary conditions permit us to extend this range, for we must have

0>(0,0 = F-ct) + G(ct) = 0 for all t. Therefore,

F(-x)=-G(x) (7)

Hence the second equation (5) permits us to define F(x) also for the range

-L ^ x g 0

Using Eq. (7), Eq. (3) can now be written

<$>(x,t) = F(x — ct) — F(— x — ct) ( 8 )

The second boundary condition for x = L now yields

0(L,*) = F(L - ct) - F ( - L - ct) = 0 (9)

or F(L-ct) = F(-L-ct) (10)

This result shows that F is a periodic function in x with the period 2L. Thus, F is defined for all values of x. Furthermore, if t increases by T = 2L/cy the function will also repeat itself. Therefore F is also defined for all times.

P[x, t)

FIGURE 3

Equation (6) has an interesting geometrical interpretation (see Fig. 3). We draw through the point P in the xyt plane the two characteristic lines PA and PB (the characteristic lines are the lines obeying the equations x = ct = constant). It is only those initial data (Cauchy data) which lie

§1.8 BOUNDARY VALUE PROBLEMS I N MATHEMATICAL PHYSICS 255

on the segment AB that contribute to the disturbance at point P at the time t. In optics the triangular polygon APB is called the retrograde light cone. If the initial velocities are zero, i.e., g = 0, then

®(xj)=%[f(x-ct)+f(x + ct)] (11)

This is an expression of Huyghens' principle: The perturbation at x at the time t, originating from the sources at A and B at the time zero, need the retardation time r== (x2— x)/c= (x — #i)/c to reach the point x.

L8* The general eigenvalue problem and the higher-dimensional wave equation. The one-dimensional wave equation is the only one for which the general solution can be given in closed form as in Eq. (3). In all other cases the method of separation of variables must be used. It will, therefore, be useful to consider those features which are common to all these problems and see the common underlying idea. We shall designate the spatial coordinates by xly x2, xz (not necessarily Cartesian) and time again by t.

The inhomogeneous wave equation is of the structure

L[u]^ p ( W 3 > - ) ^ - F ( * ! , . . . , £ ) , u = uxly...j) (1)

where L is a linear differential operator which operates on the spatial coordinates only. The linearity of L means that L has the property

L[aut + bu2] = aL[ux] + bL[u2]

where a and b are independent of x. We need also the concept of adjoint and self-adjoint linear differential

operators. Let u = u(x...i) and v = v(x...t) be any two sufficiently differentiate functions in x; M is called the adjoint of L if vL[u] — uM[v] can be written as a complete divergence, i.e., there exists a set of functions fi(xi • • • 01 fi(x\ - - • 0 > • • • s u c n t n a t

where n = dimensionality of the space, and L is called self-adjoint if L == M. We also need the concept of homogeneous and inhomogeneous boundary

conditions. We speak of a problem having a homogeneous character if, with the solution u the function cu is also a solution, where c is a constant. For this to be the case it is necessary not only that the differential equation but also that the boundary conditions, be homogeneous, e.g., u = 0 on B, or dujdn = 0 on J5, or Axu + A2dujdn) = 0 on B, where Ax and A2 are constants.


Now suppose that u satisfies a homogeneous equation but an inhomogeneous boundary condition

u= f on B.

It is then possible to continue / in such a manner into the interior of D that L[f] = g9 where g is a continuous function in D. We can now construct a new function v^f — u which satisfies the inhomogeneous differential equation L[v] = g but satisfies the homogeneous boundary conditions

The problem can be reversed in the same manner, and we can state quite generally that a homogeneous differential equation with inhomogeneous boundary conditions is equivalent to an inhomogeneous differential equation with homogeneous boundary conditions.

We shall, from now on, assume L to be self-adjoint (in the case of the wave equation, L is the Laplace operator, which is self-adjoint). If p is a function of x only, we can interpret p as the mass density, and Eq. (1) represents the dynamical equation for a continuous medium with the restoring force L[w], e.g., an elastic force and an applied, external force F. In the case of equilibrium, after the transient has died and where F is a function of x only, we obtain as a special case Poisson's equation :

The general problem to be solved is that we assume L9 p and F are defined in a certain spatial domain D. On the boundary B of D we prescribe the homogeneous boundary conditions

We want to find a solution u(xl9...9t) of Eq. (1) which satisfies these boundary conditions.

We shall first discuss the homogeneous differential equation

i.e., F s= 0. Among all the possible solutions of Eq. (3) obeying homogeneous boundary conditions, we are especially interested in the synchronous solutions, i.e., those solutions which can be written

v = 0 on B.

L[u] = F(xl9...) (2)

u(xl9...90) = <p(xl9...) and — ^ = I/J(X19...) onB

(3)

u=vixl9...9xn)g(t) (4)

§1 .8 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 257

Since we are interested in periodic phenomena in time, we assume g(t) of the form

g(t) = a cos \t -\- h sin Xt (5)

We obtain then for v the equation

L[v] + X*pv = 0 (6)

where v must now satisfy the boundary conditions imposed on u. Eq. (6) is called a Sturm-Liouville equation. We will find that if D is a finite domain, solutions obeying the boundary conditions will exist only for special values of A: Xly A2, the so-called characteristic values, or eigenvalues. The solutions belonging to them: v1(x)yv2(x)y are called the characteristic functions or eigenfunctions. The solution of Eq. (3) is given by

u = (a cos Xt -\- b sin Xt)v(xly...yxn) (7)

An eigenvalue A can have several eigenfunctions uia)

y u^K It is then called a k-fold degenerate. The eigenfunctions can be chosen in such a way that they form a " complete orthonormal " set, i.e.,

jDpUiUkdr==8ik (8)

Every function h(xly...) which has the degree of differentiability required by the operator L and obeys the required boundary conditions can be written in a generalized Fourier series

h=%cnvnx) (9)

where the constant coefficients cn are given by

cn = jD phvjr (10)

The complete, time-dependent solution of Eq. (3) is found by superposition of the particular solutions

u = S ( a n cos Xt + bn smXt)vn(x) ( H ) where

"n= jD p<pvndry bn = Y jD pxjsvjr (12)

The inhomogeneous Eq. (1) is solved by quite similar methods. But before discussing these we must introduce the concept of homogeneous and inhomogeneous boundary conditions. We speak of a problem having a


homogeneous character if with u the function cu is also a solution, where c is a constant. For this to be the case it is necessary not only that the differential equation but also that the boundary conditions be homogeneous, e.g., u = 0 on B and dujdn = 0 on B, or Au + B dujdn = 0 on B, where A and B are constants.

Now suppose that u satisfies an inhomogeneous boundary condition, e.g., u = f on B, but obeys the homogeneous differential equation Lfu] = 0. It is then possible to continue the values of u on B in such a manner into the interior of D that L[f] = g in D, where g is a continuous function in D. If we wish now to solve the inhomogeneous Eq. (1), we can without loss of generality assume homogeneous boundary conditions. We expandF(x l 9 . . . ,£) in terms of the orthonormal functions vn :

F(Xl,...,t)=%Fn(t)vn(x) (13)

n=l

and similarly 00 «(*!,...,*) = X yn('K(*) (14)

n=l If these values are substituted into Eq. (1), we obtain the differential equation for yn(t):

^ + K 2 V n = - F n (15)

A particular solution of Eq. (15) is

Yn = ^j[*nK(t-T)Fn(T)dT (16)

Using a series expansion (14) with the coefficients of Eq. (16), we obtain a solution of Eq. (1) and, by a linear superposition, we obtain a solution of Eq. (1) satisfying the prescribed homogeneous boundary conditions.

There exists a close connection between the eigenvalue problem and the calculus of variations. We shall for simplicity state this relation for the one-dimensional case.

Suppose we want to find the function (or functions) u(x) which give the integral

lba[A^hBx)u2dx=0 (17)

a stationary value. The function u is subject to the auxiliary condition

pu2dx = constant, [p(x) > 0 for a ^ x b] (18)

B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 259

and shall also satisfy the homogeneous boundary conditions

x(a):=x(b) = 0 (19)

The Euler-Lagrange equation belonging to this problem is the Sturm-Liouville equation

dx \ dx] Bu -\-Xpu=0 (20)

The stationary values of the integral Eq. (17) are achieved if u is one of the characteristic functions with the characteristic value A. This property of the characteristic values is utilized in the so-called direct methods of the calculus of variations to find the numerical value of the lowest eigenvalue and with it also the eigenfunction itself.

We shall state now the synchronous wave equation in a few typical coordinate systems and its particular solutions. The wave equation is in all cases

V2<t> + A 2 0 == 0 (21)

a. Cartesian coordinates, two-dimensional case :

a2® a2®

Particular solutions O a / 8 = e±u*x+pv) (23)

where a 2 + jS2 = A2 (plane wave). O 0 = (Ax + B)eiXy (24)

b . Cartesian coordinates, three-dimensional case : d2(b 3*(b 32

Particular solutions ®k,i,m = e i k x + l y + m z ) (26)

where k2 + I2 + m2 = A2 (plane wave).

c. Cylindrical coordinates, two-dimensional case:

a20> l a& l a2<D ' ' W + J~¥ + 7W + m = 0 ( 2 7 )

Particular solutions

O f c / f c (Ap), (circular wave) (28)

$0=(A<p + B)J0(\P) (29)

260 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS § 1.8

= «*i(^+m?) • MpVK - *•) (31) * o,o =(A<p + B)(Cz + D)J0(\p) (32)

where Jm is the Bessel function of order m.

e. Spherical coordinates:

r* dr \ r 8r ) + r* sin2 0 00 \ S l n P 8dJ

' r2 sin2 0 By' Particular solutions

1 g 2 0 + A2<D = 0

(33)

O m ,» J=/«. + a /»(Ar>±**»P„*(cos «) (34) yr

for a modulated spherical wave, and

<*>«= — (35) for a pure spherical wave.

By using the Fourier integral it is possible to build up either spherical or cylindrical waves out of plane waves, or, conversely, plane waves out of modulated spherical or cylindrical waves.

a. Spherical wave as a superposition of plane waves : iX r(nl2)—ir~> fin z>t'Ar \

(J) v dQ s i n 0 dmgiMxsind cos<p+y sindsin(p+\z cosd\) __ i 2TTJO Jo ^ r ) K '

b. Plane wave as a superposition of modulated spherical waves :

0 = ^ X ^ y i ^ A + < i / 2 , ( A » ' ) P „ ( c o S 0) = eôose = e m ( 3 7 )

c. Plane wave as a superposition of circular waves (two-dimensional case): +00

X inJnftpyn(p= eaecow= eiXx (38)

d. Cylindrical coordinates, three-dimensional case:

a2® i ao l a2® a2®

Particular solutions (modulated cylindrical wave)

§ 1.9 BOUNDARY VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 261

1.9. Heat conduct ion equation. Let T(xyyyzyt) be the temperature distribution. In a homogeneous and isotropic medium it obeys the " equation of heat conduction "

V a2 dt i )

Here a2 = k/cp, where k = specific heat conductivity, c = specific heat, p — density. If the homogeneous body D with the boundary B is imbedded in an infinite homogeneous medium of constant temperature zero, the boundary condition has the form

^ + a r = o (2)

where a is a positive constant and the normal n is directed into the interior of D. We have here again an eigenvalue problem, and the preceeding general principles therefore apply. We write again

T=v(x,y,z)g(t) (3)

and obtain for v the eigenvalue problem V 2 ^ •+ Xv = 0 in D (4)

and to + a v = = z 0 m B (5) on

The eigenfunction belonging to A is

T=Av<rx«*t (6)

If T(xyyyzy0) is a prescribed function cp(xyyyz)y which is twice continuously differentiate and satisfies the boundary conditions, then the solution can be written in terms of the characteristic functions vlyv2y...y and their characteristic values Ar, A2, ....

CO

T(xyyyzyt) = ^cnvn(xyyyz)e-^t (1)

where cn = j-jjD <pvndxdydz (8)

If the domain in which the initial temperature cp(xyyyz) is prescribed is the entire space, the solution can be written down in closed form:

T(x,y,z,t) = ^ - ) 8 jjj+_2<p(x',y',z')^2/iaHdx'dy'dz' (9)

where p2 = (x — x')2 + (y — y')2 + (z — z')2


This solution can be interpreted in the following way. Suppose the initial temperature was zero everywhere, except in a small region around the point, where it had the constant value T 0 . Let us assume this region was a small cube of size A T . We may choose this point P 0 as the origin. The temperature is then

T(X,y,z,t)=-^ê-^l^ (10) (2a\/7Tt)3

where p 2 = x2 + y2 + z2

At the time zero there exists a sharp maximum at the origin, which gradually becomes less and less defined caused by the diffusion of the heat. The initial accumulation of heat at the origin is called a heat pole. We may therefore interpret Eq. (9) as the result of a continuous set of heat poles having the strength <p(x,y,z).

Another solution of the one-dimensional heat equation

32T_ 1 8T dx2 ~ ~ a2 dt " '

of geophysical interest is the penetration of heat into the interior of a semi-infinite, homogeneous isotropic medium if the temperature on the surface is a prescribed periodic function of time, e.g., the temperature distribution in the interior of the earth caused by the daily or annual temperature oscillations on its surface.

We shall assume the temperature at the surface x = 0 to be A cos cot. The temperature T(xj) for x > 0 (in the interior of the medium) is given by

T= ^-<I/«>VW2 x c o s _ JL ^ x j (13)

This expression shows that temperature distribution has the shape of a damped oscillation, and that the phase is continuously changing. The solution also has another remarkable property;.if we write the phase in the form

-±-^(x-aV2^t) (13)

we see that the velocity aV2w with which the disturbance travels depends on co. In other words, we have here a dispersion, which was not the case for the one-dimensional wave equation. This is also the reason why we cannot have here a general solution of the type/(# ± d).

§1.10 B O U N D A R Y VALUE PROBLEMS I N M A T H E M A T I C A L PHYSICS 263

Another significant difference between the heat and the wave equation is that the heat equation is not isotropic with respect to the direction of time. The solution for t > 0 cannot be continued to values t < 0. This is connected with the fact that heat flow is an irreversible flow and is connected with an increase in entropy.

L10* Inhomogeneous differential equations* Suppose we have density distribution p of mass in space, and we assume that p is independent of time. The potential <E> then obeys Poisson's equation

V * < D = - 4 7 r p (1)

Its solution is O(P) = I ^- drQ (2)

where rPQ is the distance between the " field point." P, where we want to compute the potential, and the " source point " Q with mass pdr.

If we assume that p is a function of space and time, we obtain the inhomogeneous wave equation for <P:

1 d2<5> ^ - - ^ = - 4 ^ (3)

One solution of this equation is

O(P) = j P(Û-Rlc) w ^ ( 4 )

where R2 = (x - £)2 + (y-rf) + (z - Q2

This solution is called the retarded potential, because the contributions to the potential at P are arising from those values of p at £)(£,77,£) which needed the time Rj c to reach P.

Bibliography

1. CHURCHILL, R. V., Fourier Series and Boundary Value Problems, McGraw-Hill Book Company, Inc., New York, 1 9 4 1 . Emphasizes the expansion in orthogonal functions.

2. COURANT, R. and HILBERT, D., Methoden der mathematischen Physik, 2d ed., Vol. 1, 1 9 3 1 ; Vol. 2, 1 9 3 7 : Julius Springer, Berlin. (Also Interscience Publishers, Inc., New York, 1 9 4 3 . )

3. FRANK, P. and VON MISES, R., Die Differential- und Integralgleichungen der Mechanik und Physik, 2d ed., Vol. 1 (mathematical part), 1 9 3 0 ; Vol. 2 (physical part), 1 9 3 5 ; F . Viehweg und Sohn, Braunschweig. (Dover reprint)

4 . WEBSTER, A. G., Partial Differential Equations of Mathematical Physics, 2d ed., G. E. Stechert & Company, New York, 1 9 3 6 . (Dover reprint)

Chapter 10

HEAT AND THERMODYNAMICS B Y P. W . B R I D G M A N

Higgins University Professor Harvard University

1 . Formulas of Thermodynamics

The derivation of the word thermodynamics, which literally means " concerned with the motion of heat," suggests that the science of thermodynamics deals with the details of the interchange of heat between bodies. As a matter of fact, traditional thermodynamics does not deal with the motion of heat at all, but with the equilibrium conditions reached after heat motion has ceased, and should, therefore, more properly be called thermostatics. The results of thermodynamics are derived from two laws : the laws of the conservation and of the degradation of energy. These two laws are, perhaps, the most sweeping generalization from experiment yet achieved. A mechanistic account of these laws can be given by the methods of statistical mechanics, for which the following chapter should be consulted.

1 . 1 . Introduct ion. The partial derivatives of thermodynamics are somewhat different from the conventional partial derivatives of mathematics in spite of their apparent identity of form. The conventional derivative (dy/dx)z implies by its form that y is a function of x and z. In the corresponding thermodynamic derivative the subscript z indicates the path along which the derivative is taken, y and x not necessarily being defined off the path. It follows that such derivatives as (dT/d / ) )^ , (drldp)w, (dQldp)r> (dWldr)v are thermodynamically meaningful, although Q and W are not possible independent variables.

The ordinary mathematical formulas for the manipulation of first derivatives apply to the path derivatives of thermodynamics. In particular,

a) 264

§1.2 HEAT AND THERMODYNAMICS 265

du), 'dx)„di)v+dy)B(du),. ( 3 )

The second path derivatives, however, in general require special treatment. In particular,

~ 7 * 2 V 1 ^ * Q l ^ * Q etc (4) 3_ dp

L2* The laws o f thermodynamics . The first law is

dE=dQ-dW (1)

This formula applies to any thermodynamic system for any infinitesimal change of its state. This formula is a definition for dE, it being assumed that dQ (heat absorbed by the system) and dW (work done by the system) have independent instrumental significance. The first law states that the dE so defined is a perfect differential in the independent variables of state. By integration E may be obtained as a function of the state variables. The E so obtained contains an arbitrary constant of integration, without significance for thermodynamics.

The second law is ds = ^ (2)

T

The second law states that for any reversible absorption of heat there is an integrating denominator r (absolute temperature) such that ds is a perfect differential in the state variables.

Applied to a Carnot engine (engine working on two isothermals and two adiabatics)

ft/ft = T I / T , ( 3 )

efficiency of Carnot engine = ^* ^ 2 = — — (4)

The two laws apply to any subsystem that can be carved out of the whole system. It follows that there are fluxes of heat q, mechanical energy wy

total energy ey and entropy s such that

- div q = dQjdt (5)

div w = dWjdt (6)

e = q — w (7)

» = f / T (8)

where now Q and W are taken for unit volume, and t denotes time.

266 HEAT A N D T H E R M O D Y N A M I C S §1.3

1 3 . The variables . The variables usually associated with thermodynamic systems are p = pressure, r = absolute temperature, v = volume, s = entropy, E = energy, G = E -\- pv — rs (Gibbs thermodynamic potential), F= E — TS (Helmholtz free energy), H = E - j - pv (total heat or enthalpy), dW = work done by the system, dQ = heat absorbed by the system. The specific heats are defined as

These variables are all to be taken in consistent units, which means in particular that heat is usually to be measured in mechanical units. The amount of matter in the system to which these quantities refer may be taken arbitrarily, subject to the demands of consistency. Thus, if v represents the volume of one gram, Cv refers conventionally to one gram of matter, but measured in mechanical units. However, if v is taken more ordinarily as the volume of that amount of substance which occupies one cubic centimeter under standard conditions, then Cv has its conventional value multiplied by the density.

1.4. One - component systems. The state is fixed by two independent variables of which p and T are a possible pair. dW= pdv (one-phase systems). In such a system any three of the first derivatives can, in general, be assigned without restriction from the laws of thermodynamics (there are many exceptions to this). Any fourth first derivative can then, in general, be expressed in terms of the chosen three with the help of the first and second laws. Normally, a thermodynamic relation involving first derivatives is a relation between any four such derivatives. There are approximately 10 1 1 such relationships. Various schemes have been proposed by which any desired one of these 10 1 1 relationships may be obtained with comparatively little mathematical manipulation *. No attempt is made to reproduce these schemes here, but in the following a few of the most frequently used relations will be given.

a) (2)

* (3)

References 1, 2, 3, 4 and 10 in the Bibliography.

§ 1.4 HEAT A N D T H E R M O D Y N A M I C S 267

l8v\ idv\ T (dvV \8p)s'\8p)T

+ CP\8r), (5)

(6)

(7)

/ M - T ( 0 D / 8 T V (8)

'~Y~(dpldv\ (9)

(8VI8T)S _ 1 (8vldr)p 1 — y (10)

T \ a r ; , (11)

(f)T = *> i f ) , " ' (12)

( - a r ) , — * 1-87). = - ' (13)

(14)

(15)

(16)

Equations (l)-(4) are the four Maxwell relations.


Equation (16) is the so-called " thermodynamic equation of state."

ju. = > (Joule-Thomson coefficient)

\dpjT r W ) v

(17)

(18)

(19)

L 5 . One-component, usually two-phase systems. The state is fixed by two independent variables of which p and r are not a possible pair, dW=pdv. A new variable appears, x, the fraction of the total mass of the system present in phase 1. There are now only two independently assignable first derivatives, of which dpjdr and Cv [= (^Ql^)v] form a convenient pair; Cv is expressible in terms of the properties of the separate phases.

ZrdT\dT)v r\dr) \dpT

(1)

The general thermodynamic relation between first derivatives is a relation between any three derivatives. In these relations the following function of * occurs frequently :

f(x, 1 — x) = x (8vA (dvA d£ [dr)p^[dp)rdr_

+ 0 - * ) [ d r j ^ [dpTdr

(2)

In addition to the following special formulas, many of the formulas of Section 1.4 also apply.

(S), ~ ~~ ^ T T 2 ^ d r ) + f(-X' 1 *\ ( 3 )

§ 1.6

/8x\

(

HEAT A N D T H E R M O D Y N A M I C S

dx dv J Q v± — v9

( ).-/> — r(dpldr)

- 1

- / ( * , 1 — *)

\ M ' / G " ( ^ 1 - ^ 2 ) [ P - T ( ^ T ) ]

\TT)

l ©! — w a

- 1 «i — Vo

/(*, l—x) + Cv + v(dp/dr)

r(dpldr)

F — " 2

dr T(V1 — v9)

f(x, !—*)+••

dp , (Clapeyron's equation)

269

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

where L is the heat absorbed by the system when it passes from phase 2 to phase 1 (latent heat).

d*r - 1 dr\ Cn - C „ (dry ' "r [dp) dp2 v1 — v2dp )

dv dr dp

dvx

(dvA _(dvA 1 [ d r ) , \8T),\

If the phase 1 is a perfect gas,

d\o%p L dr Rr2

1.6. Transitions of higher orders

dp _ A(8»/8*), <*T ~A(A*/G/>)R

(12)

(13)

(14)

(1)

v1~~v2

1

270 HEAT AND THERMODYNAMICS §1-7

Here z is any continuous thermodynamic function with discontinuous first derivatives along a line in the p,r plane. The formula gives the slope of the line of discontinuity. By specializing z, various formulas may be obtained. By identifying z successively with s and v and eliminating dpjdr, Ehrenfest's formula for a transition of the second kind is obtained :

A C » - - T [ A ( 8 ^ P ) 7 J &

1.7. Equations of state

pv = RT> (perfect gas) (1)

If p is measured in dynes/cm2, v is the volume in cm 3 of one gram molecule, and t is absolute Kelvin degrees, and R = 8.3144 X 107ergs/g-mole-deg.

(2)

(3)

+ -^j (v — b) = Rr, (van der Waals)

(p + ^ 2 " j ( © - * ) = Rr, (Berthelot)

— C

p(v — b) = Rr exp (Dieterici) (4)

p = i ^ T ( 12 " " € ) ( p + P ) - ^ (Beattie-Bridgeman) (5)

where i4 = i 4 0 ^ l - ^ , J 5 = B 0 ^ l " € =

and A0, a, BQy by and c are constants. JB C

pv — A + — + ..., (" virial equation of state " ) (6)

The coefficients A, B, etc., which are functions of temperature, are called the first, second, etc. virial coefficients.

Excepting the perfect gas equation, all these equations have the property of indicating the " critical po in t " at which vapor and liquid phases are identical. The parameters of the system at the critical point are designated by pc, r c , vc. The " reduced coordinates " of a system are defined as pr = p/pc, rr = t / t c , vr = vjvc. If the various equations of state are expressed in terms of reduced coordinates, a functional form will be obtained, the same for all substances. In particular, the reduced van der Waals equation of state is

§1.8 HEAT A N D T H E R M O D Y N A M I C S 271

1.8. One-component, two-variable systems, with dW'= Xdy* Here X is " generalized force " and y is " generalized displacement/' Possible pair of variables, r and X, The simplest examples of such systems are electrical or magnetic systems, or simple elastic systems with a single stress component not a hydrostatic pressure. All the formulas of § 1.4 apply, replacing p by X and v by y.

1.9. One-component, two-variable systems, with dW = Xdy* Here X is generalized force and y is generalized displacement, T and X not a possible pair of variables, X being fixed as a function of r by the physics of the system. The formulas of § 1.5 apply, replacing p by X and v by y. Special examples are

u— constant r 4 (Stefan's law) (1)

where u is the density of black-body radiation.

the equation for the temperature change of the emf of a reversible cell in terms of the heat absorbed when unit quantity of electricity flows through the cell isothermally.

the temperature derivative of surface tension in terms of the heat absorbed when the surface increases isothermally.

1.10. One-component, multivariable systems, with dW = 2 Xidy^ Here the X / s are generalized forces and the y / s are generalized displacements. Much variation is possible in the treatment, depending on the choice of independent variables. The generalized equation of state must be known for such systems. A general method of getting information is to put ds a perfect differential and write the relations on the cross derivatives of the coefficients. The various potential functions may be generalized for such systems. For example,

This potential function is a minimum at equilibrium for changes at constant r

(Helmholtz) (2)

(3)

GGEN = E - rs + ZXy (1)

and X.

1.11. Multicomponent, multivariable systems. dW = pdv. A system fixed by composition variables ni9 usually given in gram molecules, plus two others, for example p and r. In general a composition variable


will be needed for each independent component in each homogeneous part (phase) of the system. A complete scheme for obtaining all the possible relations in such a system has been given by Goranson (see Bibliography), analogous to the schemes for dealing with § 1.4. Only a few of the more important relations will be given here. For arbitrary values of the independent variables in multicomponent systems, the system is generally not in equilibrium, and the spontaneous changes of variable which occur as the system settles toward equilibrium are irreversible. At equilibrium

G is a minimum for changes at constant/) and T (1)

F is a minimum for changes at constant v and T (2)

i f is a minimum for changes at constant 5 and p (3)

E is a minimum for changes at constant s and v (4)

In general all the formulas of § 1.4 apply with the addition of all the n-s as subscripts. For example,

(5)

The new formulas involve derivatives with respect to the w/s. These derivatives define various " partial " quantities. In particular the " partial chemical potentials " are defined by

„_(«) _(«£) .(«) .(«) w

Here the subscript tij indicates that all the w's except «^ are held constant during the differentiation. Other typical partial quantities are

- (£) (7)

(8)

\ 8n< j C«~\^\ (9)

No subscripts are indicated in these derivatives. The understanding is that all the other independent variables except ni9 whatever they may be, are held constant during the differentiation.

§1.12 HEAT AND THERMODYNAMICS 273

For the various extensive properties of the system there are linear relations on the m's, such as

G = S n # i (10)

F^-Lnm (11)

0 = S » ^ (12)

s = 2 « A (13) Formulas for the various partial quantities hold similar to those for the

total quantities. For example,

l8M = - r

( 8P/rini

(14)

(15)

1.12. Homogeneous systems, a. Gaseous mixtures. The mole fractions are defined by

S nk

2Nt=l

The partial vapor pressures are denned by

Pi = Nip

%Pi = P

Perfect gaseous mixtures :

P

Pi V

(1)

(2)

(3)

(4)

(5)

(6)

(7)

3r ML T 2

(8)

274 HEAT A N D T H E R M O D Y N A M I C S § 1.12

dr RT2

where Lv0 is the heat of reaction at infinite volume.

(16)

b. Ideal solutions : Pi^lLiO + RrlogNi (17)

where /x*0 is a function of pressure and temperature but not of composition. 3^ _ dpf-dp dp

dr dr

vt (18)

(19) r

Ideal solutions of the same solvent mix with no change in volume or total heat content.

If the perfect gases react according to the equation

IL + mM ...<±qQ+ rR ... (9)

where L, M , etc. represent moles of the reacting species, and /, m\ etc. are the numerical coefficients in the reaction equation,

^°)ir-*\m''' = K(r)> ( m ass action law for perfect gases) (.10) (PL)(PM) •••

Here K(r) is a function of temperature only and is called the equilibrium constant. " ,. r

dr ~ ~ RT* K

where Lp is the latent heat at constant pressure of the reaction from left to right.

Imperfect gaseous mixtures: The " fugacity " (== pt*) of the ith component is a quantity closely related to the partial vapor pressure and is defined by the equations

^ = ^ 0 ( T ) + i ? r l o g ^ * (12)

l i m i t s * = Pt (13)

\~ Br V I N - RT* ( L V

where /x ° and are the limiting values at infinite volume. If the imperfect gases react according to Eq. (9)

( p L * y ( P M * r . . . - K T ) ( 1 5 )

§1.13 HEAT A N D T H E R M O D Y N A M I C S 275

L I 3 . Heterogeneous systems. The phase rule of Gibbs is

f=c-n + 2 (1)

where / is the number of degrees of freedom (counting intensive variables only), c the number of components, and n the number of phases.

For a liquid or solid solution in equilibrium with a vapor which behaves as a perfect gas

E A ^ ( l o g & ) = 0» (Duhem-Margules) (2)

for changes in composition at constant p and r, where Nt refers to the condensed phase and p to the vapor phase in equilibrium with it.

pi = ktNi9 (Henry's law for ideal solutions) (3)

where kt is independent of N.

pQ = pQ°N0l (Raoult's law for ideal solutions) (4)

Here the subscript 0 denotes the solvent, and j>0° the vapor pressure of the pure solvent

11 = log -^r- = Nly (osmotic pressure) (5)

Here II is the osmotic pressure of the ideal solution of the component Nx

in the solvent N0. Depression of the freezing point of an ideal dilute solution

0 = ^ ! ^ , (van'tHoff) (6)

where Lf° is the latent heat of freezing of the pure solvent. Raising of the boiling point of the ideal dilute solution

p T 2 e = inrNi> (van'tHoff ) ( 7 )

where Lv° is the latent heat of vaporization of the pure solvent at the normal boiling point r & ;

N-* —~ = K, (Nernst's distribution law for ideal solutions) (8) is i

where oc and jS denote different condensed phases, and K depends on solvent, p, and r, but not on Nt.


Bibliography

1. BRIDGMAN, P. W., Phys, Rev., April, 1914. 2. BRIDGMAN, P. W., A Condensed Collection of Thermodynamic Formulas, Harvard

University Press, Cambridge, Mass., 1925. 3. CRAWFORD, F. H., Am. J. Phys., 17, 1-5 (1949). 4. CRAWFORD, F. H., Proc. Am. Acad. Arts ScL, 78, 165-184 (1950). 5. EPSTEIN, P. S., Textbook of Thermodynamics, John Wiley & Sons, Inc., New York,

1937. A comprehensive book for the more advanced student. 6. GORANSON, R . W., Publication 408, Carnegie Institution of Washington, Washing

ton, 1930. 7. GUGGENHEIM, E . A., Thermodynamics, North Holland Publishing Company,

Amsterdam, 1949. Features especially the use of thermodynamic potentials. 8. PRIGOGINE, I. and DEFAY, R . , Thermodynamique chimique, Editions Desoer, Liege,

1950. Deals with " open " systems and irreversible processes. 9. SCHOTTKY, W., Thermodynamik, Julius Springer, Berlin, 1929. Notable for

rigorous examination of fundamentals. 10. SHAW, N., Trans. Roy. Soc. (London), 1935, pp. 234, 299. 11. ZEMANSKY, M. W., Heat and Thermodynamics, 3d ed., McGraw-Hill Book

Company, Inc., New York, 1951. A comprehensive treatment from the elementary point of view.

Chapter 11

STATISTICAL MECHANICS B Y D O N A L D H. M E N Z E L

Professor of Astrophysics at Harvard College Director of Harvard College Observatory

1. Statistics of Molecular Assemblies

1.1. Partition functions. Thermodynamics and statistical mechanics constitute independent approaches to essentially the same problem : the physical properties of assemblies of atoms or molecules. Thermodynamics depends on certain postulates concerning the behavior of an artificially defined quantity called entropy. Statistical mechanics concerns itself with averages over an assembly wherein a certain amount of energy E has been partitioned among the various atomic or molecular systems.

Fowler defines a quantity known as a "partition function/' for each component of the assembly. By component we mean each kind of atom, molecule, particle, or " system," whether interacting or not. Thus all atoms of neutral iron comprise one component; those of singly ionized iron a second, etc. In its most general form the partition function, f(T), reduces to a sum :

f(T)^=ê-^T (1) i

where ^ is the energy of state iy k is Boltzmann's constant, and T is the absolute temperature. If a given level is degenerate, consisting of states, we sum by multiplying the summand by <x>h which we term the statistical weight.

For a molecule, we often can break up the total energy of a given state into a number of independent energies, viz., e v = potential energy, € T = kinetic energy, e r = rotational energy, ev = vibrational energy, ee = internal electronic energy.

Of these energies the last three are usually quantized and, for classical assemblies, the first two are nonquantized. The potential energy is a function of the coordinates ql9 q2) q%; the kinetic energy, in turn, depends on the momenta ply p2, />3.

277

278 STATISTICAL MECHANICS §1.1

Writing € == €y£- €T+ €r + €v + €e (2)

we see that the partition function factors whenever the energies are independent as above. Then

f(T) = / / / W « V M M f c JJJ e ^ r

r v e

= V(T)H(T)Br(T)Bv(T)Be(T)

The quantity A is Planck's constant, which we include so that the product dp^dqjh will be dimensionless. We arbitrarily associate h with the kinetic rather than with the potential energy.

V(T) = jjj e-°vl*Tdqidqidq3 = V (4)

the volume, in the absence of an external potential field. Under a constant gravitational acceleration g> the potential is e v = mgz> where m is the mass of the particle and z the vertical coordinate. Then

V(T)= e~m9ZlkTV (5)

Usually we integrate over only a small volume V of the assembly, wherein we can take e v constant or zero.

The classical partition function for the kinetic energy becomes

A factor H^z belongs to each degree of freedom of kinetic energy. For a diatomic molecule, we can write

cr=hcBJU+V) (7)

where c is the velocity of light, B the quantity

B = hl%7T2d (8)

with I the moment of inertia, and / the quantum number, an integer : 1,2,.... For a given / , we get 2 / + 1 states of identical energy. Thus

BAT)** Û+le-J(Ul)hcBllcT (9)

§ 1 . 1 STATISTICAL MECHANICS 279

When hcB/kt >* T we cannot easily simplify the expression. But when this quantity is small, we can approximate to Br( T) by an integral

Br(T)~ 1 e-JiJ+DhcBlkTJJ

(10) 1 kT ^JifilkT

Ba he Wo

We call a the symmetry factor. It is unity for diatomic molecules whose components are unlike, e.g., C 1 2 - C 1 3 . It is 2 for diatomic molecules whose constituents are identical, e.g., C 1 2 - C 1 2 . The number of rotational states is halved for a symmetrical molecule.

The spacing of the vibrational levels, if independent of rotation, will usually follow a law similar to

ev = (v + ±)hcoje (11)

where a)e is a basic vibrational wave number and v an integral vibrational quantum number.

Equation (11) measures ev from the lowest point of the potential energy curve. If, instead, we measure it from the condition of complete dissociation, we get

ev= ^v + ^hcoje — D — ~Y = hcaje^ — D (12)

with D equal to the dissociation energy. Then oo DjkT

B * T ) = t e ° - » » > . * l * T = eJ t T - (13)

0 1 C

We cannot usually express the electronic energy in any simple form. We must therefore merely write

BeT)=^&ie-^T (14) i

with the sum taken over all relevant levels of electronic excitation. For atoms we take the statistical weight of a single state to be unity. If, as in common practice, we represent a level by its inner quantum number, / ,

<3,= 2 / + l (15)

If a complete term, denoted by the orbital and spin quantum numbers L and 5, is degenerate,

etc. (see Chapter 19). Wj = (2.S + 1) (2L + 1) (16)

280 STATISTICAL MECHANICS § 1.2

A spinning electron has the invariant weight:

= 2 (17)

For molecules we encounter electronic states characterized by the quantum numbers S and A . A = 0 corresponds to 2 levels; A = 1 to I I ; A = 2 to A, etc. If we set r = 2S + 1, where r is the multiplicity, we get

co, = 2S + 1 = r for r S levels \

ioj = 2(2S + 1) =• 2r, for *TI, r A , etc. levels | ^

The complete partition function for a diatomic molecule whose components possess masses m1 and m2 becomes

P V F ) ~B~o ~hc (\— e-^Tj BlT) (19)

L 2 . Equations o f state* For a perfect gas, the equation of state becomes

p = ^.l^ = N k T J _ l n f T ) ( 1 )

Since V usually enters i n t o / ( T ) only as a multiplicative parameter, we get

p = ykT=nkT (2)

where n is the number of systems per unit volume. If we take V as the volume of a gram molecule and set N equal to the number of atoms per mol,

pV=NkT=R0T (3) If p is the density,

p = pkT\m^ (4)

where m is the mass of an atom of unit atomic weight, and p, the atomic weight of the gas.

For an imperfect gas the best-known equation is that of van der Waals :

(p + alV2)(V-b) = NkT (5)

Berthelot gives an alternative empirical equation :

(p + a'I TV*) (V - b) = NkT (6)

and Dieterici still another, viz :

p(V-b) = NkTe-a'/NkT8V (7)

where a, a\ b, and s are all constants, the last being an exponent.

§ 1.3 STATISTICAL MECHANICS 281

(3)

where the subscripts indicate that V and p are held constant during the respective differentiations; M 0 is the number of particles per mol.

1.4. Adiabatic processes. Let y be the ratio

Y=CJCy (1)

The so-called adiabatic law, wherein no heat is added or subtracted from a gas is

pVv = constant (2)

pTv/a-y) = constant (3)

p yi/ci-y) = constant (4) In each of these equations we regard V as the volume of a mol (or some other conveniently defined mass of gas.)

T o the first order in \\V these three equations assume the respective forms,

p V = N k T + iL^=^ ( 8 )

p V = N k T ^ N k T b - a ' ^ (9)

P V = N k T ^ N k T b - a ' l T ^ (10)

1.3. Energies and specific heats of a one-component assembly. If E is the total energy assigned to the N systems of a one-component assembly,

E^NkT2~lnf(T) (1)

The mean kinetic energy per particle is

€ = ^ = ^ A i N / / ( T ) = ^-^R (2)

The specific heats per mol at constant volume Cv, and at constant pressure Cv are


£ = £ i V , . W ^ l n / , ( r ) (1)

1.5. Maxwell's and Boltzmann's laws* The number of systems in a given quantum state is

1 B(T) 1 ' which is Boltzmann's law. Also

A = ^ r « r » i ) / * r (2) Nj coj v '

In a classical assembly, the number, <f> of systems in a volume element dV> with components of momenta in dpx, dp2, dps, becomes

/ (3)

— . 1 />-<j>i8+j>«2+i>»8>/*r//ft /i* rf* I (27rmAT)ê apldp2apsav j

normalized to give

l^dpjpJV = NIY (4)

where the integral is over a unit volume, with the respective momenta limits —oo tO +co.

In terms of Cartesian velocity components, w, v, w, we have the alternative form of Maxwell's equation :

/V7 m \3/2 cf>dudvdwdV = y i ^ f ) e-m(u2+v2+^l2kTdudvdwdV (5)

If r is the actual space velocity, such that

c2=u2 + v2 + w2 (6)

the number of systems moving with velocities in dc and within solid angle dw are

/<fc<fa> = ~ I c2e-™*l™Tdcdw (7)

T o get the total number per cm 3 , with velocities in dc, replace dco by 4TT. For additional theorems on kinetic theory of gases see Chapter 20, pp. 288-

302.

1.6. Compound and dissociating assemblies. When an assembly consists of more than one component, interacting or not, we write, instead o f E q . ( l ) o f §1.3,

§ 1 . 7 STATISTICAL MECHANICS 283

where the subscript j refers to a given component. The partition function is the same as before. For noninteracting components Nj is constant. Otherwise, will depend on T and V according to the dissociation equation. For example, when Nm atoms of component m interact with Nn atoms of component n to form Ns compound molecules of component s, in accord with the symbolic chemical equation

M + N+±S (2)

where S is a molecule of type MAT, the equilibrium conforms to the law

NmNn fJTtfJT)

In a more general case, when Nm, Nn) Npy etc., interact to form a compound molecule, so that oc atoms of type my j8 of type « , y of type p, etc., are required to form /x compound molecules of type s, according to the reaction

a M + |3iV + yP + ... = y,S = F [ M A / ^ / A , P W M . . . ] (4)

in ordinary chemical notation. The general dissociation equation becomes

Nm*Nj>N,y... _ (fmHfny(fPy... Ns" ifsy ~ ( 0 )

in which the superscripts a, j8, y, \i are true algebraic exponents. For more detailed forms of these equations, including the Saha " ionization formula," see Chapter 21.

1.7. Vapor pressure. The vapor pressure of a gas in equilibrium with its solid or liquid phase is

* k T J(T) n \ p = - v - - - ( f ) ( 1 )

For structureless systems, the partition function

fT) = V™*pmVe-zl*T (2)

where % is the energy required to evaporate an atom from the liquid or solid to the gaseous phase at absolute zero. The partition function for the crystal is

l n K ( T ) = jTJV-£LjTQC]>)soldT' (3)

where (Cv)soi is the specific heat of the solid or condensed phase, calculated at the vapor pressure of the gas for the value of T in the integrand; (Cj)^ also includes heats of transition (melting, evaporation, etc.)

284 STATISTICAL MECHANICS §1.8

For molecular gases or other gases having internal structure, the appropriate factors relation to rotations, vibrations, electronic energies, etc., must be included in f(T). For many applications we can set, approximately,

K(T)~H(T) (4)

see Eq. (6) of § 1.1. More generally we note that the empirical relation,

p = CT«e-xlT (5)

with three disposable constants, C, a, and X usually gives a very accurate representation between any two given transition points.

1.8* Convergence of partition functions. The partition function Be(T)y Eq. (14) of § 1.1, does not converge if we sum over all quantum numbers to n = » . The presence of neighboring atoms fixes an effective upper limit to the summation, beyond which we regard the electron as free rather than bound. Thus we excluded the volume assigned to other atoms. Denote by the subscript s the lower electronic states of a given atom, and by r the lower electron states of the same atom in the next higher ionization stage. Approximately

s

The bracketed constant, we shall term Q. Its value is

0 ~ 1 . O 2 X 1 0 1 2 c m " 3 / 2 (2)

In this equation, coe is the electron spin, Z is the effective charge of the higher ionization state involved, Xo is the ionization potential, and n0 is the number of systems per cm 3 . Usually only the first or the second term of (1) will dominate.

1.9. Fermi-Dirac and Bose-Einstein statistics. The foregoing formulas, especially those relating to equations of state and partition of kinetic energy, involve the tacit assumption that the energies of such a system are not quantized. This assumption is sufficiently correct at low densities and high temperatures. However, at high densities and low temperatures, we find that even the so-called " continuous states " are quantized and, furthermore, that we can assign only one particle to each state. The Pauli exclusion principle is a consequence of this general law.

64/_3_VV2 ( J t t c f 9 7 r \ 4 7 7 / € 3

§ 1-9 STATISTICAL MECHANICS 285

For systems whose wave functions are antisymmetric, we get the Fermi-Dirac statistics (indicated by the upper sign in the following formulas). When the wave functions are symmetric, we get the Bose-Einstein statistics (lower signature).

In our given volume V, which contains N systems in all, we find that Ns

are assigned to the energy level es, whose statistical weight is 6js. Then

i V s = ± ^ | l n ( l ± A ^ r ) = F _ ^ T ( 1 )

The parameter A follows from the condition that

XiV 5 =iV=XA-V^ ± 1 (2) For Bose-Einstein statistics we must have

0 < A < 1 (3)

For Fermi-Dirac, we have the condition

A > 0 (4)

T o complete these equations we must have formulas for the separations es. For kinetic energy in a cubical volume bounded by a side of length /,

where x, y, and z are integers 0 < x9y,z < » . Then

OO 00 00 ~

N = S S S \-leh*(a*+v*+z*)./6l*mkT ± 1 ( 6 )

where a> is a weight factor set equal to unity for longitudinal and equal to 2 for transverse waves. The quantity h2l8l2mkT = 4.4 X l O - 1 1 , if / = 1 cm and T= 1° K. Hence we can approximate to the sum by an integral

wdxdydz J 0 J O J 0 X ! \-lehHx2+y2+z*)/8l2mkT _j_ 1

Mi Mi rco cor2 sin 6 drdddcp | J 0 J 0 Jo ^-iehyi8l2mkT ^_ I ('

_ _77_ roo &r2dr _ 277co(2m)3/2F r«> e / / 2

~ 2 J O X-.lehr*/8l*mkT ±~l ~~~ & Jo A - V . / * ^ 1 € «


N L _ ^ _ ^ (15) 3=1 J

I AO* Relativistic degeneracy. At extremely high temperatures we must allow for relativistic change of mass with velocity. The deBroglie wavelength of a particle moving with velocity v is

where m0 is the " rest mass." The kinetic energy is \-l/2 [(-?r i (2)

o r 1 _ 2e,m 0 (l + e s /2m 0 c 2 ) A„2 W- (3)

where we set x2+y2 + z2 = r 2 (8)

and transform from Cartesian to spherical coordinates. We have se t / 8 = V. We cannot integrate again in finite terms. However, when A 1,

N ~ h* J o «-e</* V a * a = ( 9 ) Thus, when

\ N h * ^ 1 MAX

a form of Maxwell's law, expressed in terms of energies instead of momenta or velocities as before. Thus, when A << 1 we recover the classical formulas.

When A 1 (for the Fermi-Dirac case only)

where e' is defined by A - V ' / f c T = l (13)

For large A, then, the Fermi-Dirac distribution becomes Ns = 2TTto2mfjWes

1,2des (14)

for the energy range below e . For higher energies, the formula grades into the classical expression.

For the Bose-Einstein case, A < 1, we get

§ 1.11 STATISTICAL MECHANICS 287

N =

When A < 1,

When A > 1,

~W*~ Jo A - W f c T + 1 s W

N~8TTGJVA^J (7)

„ Air&V/kT. ,\3

1.11. Dissociation laws for new statistics. Let

Pm = nijme (1)

the ratio of the mass of the system to that of the electron. For each component m of an assembly find the appropriate value of Xm

according to the conditions of temperature and density.

Classical nonrelativistic: \ Nm h3

amV (2TTmmkTfl* w

\ NJV<&Jh\2-nmmkTflÂ.%l X l O ^ T ) ^ ) conditions: m „ >

( T<mmc2/k = 5.9 X 1 0 9 £ w j

Classical, relativistic:

&7TWmV

\ NJV<4.S1 X l O ^ T ) 3 / 2

conditions: _ „ ( T > 5.9 X l O 9 ^

Fermi-Dirac degeneracy, nonrelativistic:

h2

In A,

conditions :

he \3

A T m / F > 4 . 8 7 X K P ^ r ) 3 ' 2

T < 5.9 X 1 0 9 £ m

/ 3 N \2/3

S K - R ( 4 )

when relativity effects are predominant,

e s > 2 m 0 c 2 (4)

he and €s=j^=hvs (5) Planck's relation. The energy is independent of m0.

For Fermi-Dirac statistics we get, for a relativistically degenerate gas,

288 s t a t i s t i c a l m e c h a n i c s §1.12

3=1 J (5) '3/2 comV 2rrmmkTff*

Conditions : same as for Fermi-Dirac, above.

Fermi-Dirac degeneracy, relativistic (electron gas) :

\( NJV > (kT/hcf (477(3J3) = 2.86T3

conditions: m „ „ x

( T > 5.9 X 109/ln A w

These equations ignore internal excitation energies. Now in a reaction such as

M + N*±S ( 7 )

where S is a molecule of type MN, of dissociation energy Qsy we can determine the equilibrium from the equation

(8)

For the more complex reaction

ocM + pN+>yS (9)

with d, jS, and y integers, in which the chemical formula for S is M^YN^yy

we get A / - A m « A w W ^ (10)

L I 2 . Pressure of a degenerate gas. The general equation of state still holds for nonrelativistic statistics :

/» = T - - p r ( 1 )

£km = p J o F W ^ d T T 8 ( )

For small A, we recover the classical expression

p = NkTjV (3)

For A >> 1, for Fermi-Dirac statistics, _ _ 4 7 r ^ _ / _ 3 _ N V'3 2TT3 m(kTf( 3 i V W 3 ,

^ " 1 5 m \4m3* F / 9 /*2 \4*rf3 7 / + " ' )

Note that the leading term gives a pressure independent of T.

Bose-Einstein degeneracy, nonrelativistic:

§1.13 STATISTICAL MECHANICS 289

For relativistic statistics,

E k l n = = l ^ U A - W ? r ± 1 8 ( 7 )

When X< 1, = JVfcT/F (8)

When A >> 1, Fermi-Dirac,

^ = T ( 4 ^ ) ( F ) ( 9 )

1.13. Statistics o f light quanta. Quanta obey the Bose-Einstein relativistic statistics with A = 1. This condition is equivalent to the assumption that total number of quanta are not conserved.

E ^ ~ ^ - ] 0 e h v j k T _ _ x d v (1)

Set a> = 2 for transverse waves. Then the energy density of v radiation becomes

8TT7*V3 1 , . P > = - g T - g »» ;*r - 1 d v ( 2 )

For additional radiation formulas, see Chapter 21.

Bibliography

1. CHANDRASEKHAR, S. , Monthly Notices Roy. Astronom. Soc, 95, 2 2 5 ( 1 9 3 5 ) . 2 . FOWLER, R . H . and GUGGENHEIM, E . A . , Statistical Mechanics, Cambridge Uni

versity Press, London, 1 9 3 9 .

For Bose-Einstein statistics

„ (ItrmflHkT)^ A \> P = * hz ZfpH (5)

Chapter 12

KINETIC THEORY OF GASES : VISCOSITY, THERMAL CONDUCTION, AND DIFFUSION

The kinetic theory of gases was the pioneer branch of statistical mechanics, applied to the motion of gas molecules moving freely except during the brief fraction of time occupied by collisions. Kinetic theory thus provides the necessary connection between the microscopic viewpoint of the molecules on one hand and the macroscopic viewpoint of hydrodynamics on the other. The formulas here given were chosen as those most likely to be useful to the working physicist, especially those who like to know the steps leading the mathematicians to the results as well as the results themselves. Some of the following formulas, as, for example, those for diffusion, are not so generally well known as they should be.

1. Preliminary Definitions and Equations for a Mixed Gas. Not in Equilibrium

1.1. r,x,y,z,t, dr* Position is indicated by the position vector r or by orthogonal coordinates x, y, z; and time by t. A small volume element enclosing the point r is denoted by dr.

1.2. m, 7z, p, w 1 0 , n12, m0. A mixed gas.of two constituents is considered; suffixes 1,2 refer to the two constituents, e.g., ml9 m2 denote the molecular masses, nly n2 the number densities (or numbers per cc), p l y p 2 the mass densities n^nx, n2m2. The total number or mass density is denoted by n or p.

B Y S Y D N E Y C H A P M A N

Oxford University

Also let

so that n10 + n20 = 1

and "let m0 = mx + m^ m = pln

so that m0 is the combined, and m the mean, molecular mass.

n~nx + n2, p = p x + p 2 = nxmx + n2m2

n 1 0 == njn, n20 = n2\n, n12 = n^\n2

(1)

(2)

(3)

(4)

290

§1.3 KINETIC THEORY OF GASES 291

Thus the suffix 1 or 2 refers to one constituent only, and without the suffix the reference is to the whole gas (or to a simple gas); this is here called the suffix convention and is to be applied in each subsequent section concluded by (S.C.) meaning suffix convention.

1.3. The external forces F, X, Y, Z,W. Every molecule is supposed to be subject to an external force mF, with components mX, mY, mZ, the same for all those of one kind (tn1F1 or m2F2). Moreover it is supposed that these are conservative forces with potentials F^ *¥2, so that

grad F 2 = - g r a d Y 2 , (S.C.) (1)

1.4. c, u, v, w, dc. The velocity of a typical molecule is denoted by c, and its components by w, v, w; these may be regarded as specifying a point in a velocity space, referring to molecules of one kind only, if suffix 1 or 2 is added, otherwise to molecules of both kinds (or to those of a simple gas). A small volume element in the velocity space, enclosing the velocity point c, is denoted by dc. -(S.C.)

1.5. The velocity distribution function / . Each of the ndr molecules in the space volume element dr at time t has its own velocity point; the number of these within the velocity volume element dc is denoted by fdrdc, that is, fdr is the number density at c in the velocity space; / is a function of c, r, t in general, and to indicate this it is also written f(c,r,t); it is called the velocity distribution function. (S .C)

1.6. Mean values of velocity functions. The average of any function F(c), vector or scalar, e.g., c or u or uv or c2, over all the molecules (of either or both kinds) in dr at r, t, is indicated by the function symbol with an overline, e.g., c, F(c), u, uv, c. Clearly

nF^) = $Fc)fc,r,t)dc (1)

the integral being taken over the whole velocity space. (S.C.) In consequence of this definition,

nc = nxc± + n2c2, nu = + n2u2, etc. (2)

1.7. The mean mass velocity c0;u0,v0,w0. The mean momentum

per molecule, for the whole gas, is n-^tn^ + n2m2c2)/n; this is denoted by mc0; and u0, v0, w0 denote the components of c0. Clearly

pCo = piCi + P2C2 (1)

292 K I N E T I C THEORY OF GASES § 1 . 8

1.8. The random velocity C ; U, V, W\ dC. The random velocity C of a typical molecule, and its components Uy V, W, are defined by

C = c — c0, L7 = w — z/0, etc. (1) Clearly

Ci — cx Co, ^2 ~ 2 ô> C = c CQ (2) If the origin in the velocity space is moved to the velocity point c0, the

velocity position vector of a typical molecule is changed to C, and the distribution function / can be regarded alternatively as a function of C, r, t, namely f(C,r,t); if denotes a velocity volume element enclosing the velocity point G, the equation for F(c) in § 1.5 can alternatively be written

nF^) = lFc)fCî)dC, (S.C.) (3)

1.9. The " heat " energy of a molecule £ : its mean value E. The heat energy E of a molecule is defined to be the energy |mC 2 of its random translatory energy, together with any additional energy, either rotatory or internal (e.g., vibratory), which is communicable from one molecule to another on encounter', for example, this would not include any rotatory energy of a smooth, rigid, elastic, spherical molecule. Its value at r, t is denoted by E. Naturally E is a function of the (absolute or Kelvin) temperature T.

1.10. The molecular weight W and the constants mw NL. A molecule of mass m is said to have molecular " weight " W equal to m\mu, where mu ( = 1.6603 X10" 2 4 gram), the unit atomic mass, is one-sixteenth the mass of the O 1 6 isotope of oxygen. A mass of W grams of a gas of (mean) molecular weight W is called a gram molecule or mole; it contains NL

molecules, where NL == l/mu = 6.023 X 10 2 3 (1)

NL is called Loschmidt's number (or sometimes, less appropriately, Avogadro's number; see § 2.3).

1.11. The constants / , A, R. The following constants are used in later sections, e.g., 1.12.

/ = 4.185 X 107 ergs per calorie (Joule's mechanical equivalent of heat). k = 1.3805 X 1 0 - 1 6 erg per degree C (Boltzmann's constant). R = kNL = 8.314 X 107 ergs per degree C per mole (the gas constant in

mechanical units). R = kNL/f = 1.9865 calorie per degree C per mole (the gas constant in

heat units).

§ 1 . 1 2 KINETIC THEORY OF GASES 293

1.12. The kinetic theory temperature T is defined by

kT (1)

whether or not the gas be in thermodynamic equilibrium (see §2.3).

1.13. The symbols cv, Cv denote the specific heat at constant volume, per gram (cv) or per mole (Cv): that is, the heat required to raise the temperature of this mass of gas by 1° C, at constant volume; this is indicated formally by

cv = (1/m) dEjdT)v, Cv = We, = ( 1 R ) dEjdT)v = NL(dE/dT)v (1)

If the molecules have no communicable rotatory or internal energy, E = -|mC2 = fkT, and cv= 3kl2m; and Cv—3kj2mu, the same for all gases. If E is a constant multiple (s) of \mC2, so that E = fskT, cv = 3ks/2m, Cv = 3k$j2mu; for many diatomic molecules at ordinary temperatures s = |> as they possess rotatory energy of amount two-thirds the random translatory kinetic energy (see §2.4 for Cpy y) .

1.14. The stress distribution.p x x ,p x y y . . . . The x,y, z components of the stress (force per unit area) exerted at r, t, upon the gas on the positive side of a plane through r normal to Ox, by the gas on the negative side, are denoted by pxx, pxy, pxz; similarly p pyz, and pzx, pzy, pzz denote the stress components across planes normal to Oy and Oz. These nine components are the elements of the stress tensor; pxx, pyy, pzz are normal stresses; the other six are tangential stresses.

In the interior of a gas of ordinary low density (e.g., normal air) these stresses arise mainly from the transfer of momentum by molecules crossing the planes concerned; this produces the kinetic stresses. In addition there is a much smaller part due to intermolecular forces; a gas in which this part is negligible is called a perfect gas. The gas here considered will be supposed perfect, but the equation of state will be given also for a slightly imperfect

In a perfect mixed gas the constituents contribute independently to the stresses :

gas (§2.5).

Pxx — (Pxx)l + Pxx)& Pxy —

(Pxx)l= Pl^l2> (Pxy)l = (Pxy)i= piUxV^ etc. Pxy — (Pxy)l + PxyJ2> e*c-

(1)

Hence (see § 1.6)

Pxx ~~ = PiUf + PJJ? = plP, pxy=PUV, etc. (2)

294 KINETIC THEORY OF GASES §1.15

1.15. The hydrostatic pressure p\ the partial pressures ply p2. The hydrostatic pressure p is defined as the mean of the three normal stresses ; that is,

P = kiPxx + Pvv + Pzz) = \pC2 = kflT (1)

using § 1.12. The " partial " pressures ply p% caused by the separate constituents are similarly given by

/ > i = 3 P I c i 2 > P*=*PiP%> P^Pi+Pz (2)

1.16. Boltzmann's equation for / . Boltzmann showed that each velocity distribution function fly / 2 , satisfies an integro-differential equation. This (e.g., forfj) equates the following combination of partial derivatives offly

to a multiple integral (not given here), which involves both f± and / 2 , and represents the rate of change of fx caused by molecular encounters. In a perfect gas only simple encounters, each involving but two molecules, need be considered. The evaluation of this integral naturally, in general, involves knowledge as to the properties of the molecules, that is, as to their modes of interaction in encounters.

1.17. Summational invariants. Certain properties are unchanged by an encounter. These are (1) the number of molecules involved, namely two, (2) their combined momentum, namely mxcx + m-flx'y m2c2 + m2c2 y

or mxcx + #*2C2> according as the encounter is between like or unlike molecules, and (3) their combined energy, E + E\ or \mC2 + \m'C'2 (suffix convention) if the molecules have no rotatory or internal energy. These properties (number, mcy and E) are called summational invariants of an encounter.

1.18. Boltzmann's H theorem. For a uniform gas whose molecules are spherical and possess only translatory kinetic energy, and are subject to no external forces, the ' differential ' side of Boltzmann's equation reduces to dfx\dt. Boltzmann also considered in this case the variation of the function H defined by J / l n / ^ c , integrated over the whole velocity space. He showed that dH/dt is negative or zero, and that in the steady state, when it must be zero, In / must be a summational invariant, and consequently (see §1.16) of the form ax-\- a. tnc + a^'mC2, where aly ay a / ' are arbitrary (scalar or vector) constants; the middle term is the scalar product of di and mc.


2. Results for a Gas in Equilibrium*

2.1. Maxwell's steady-state solutions. Maxwell determined / in three important special cases, of steady or quasi-steady states; Boltzmann improved the proofs by means of his H theorem (§1.18). Later he showed that H0, the space integral of H over a given volume of gas in such a state, satisfies the equation

where S denotes the entropy of the gas. In all three of Maxwell's special states,

which is known as the Maxwellian velocity distribution. His three cases are :

a. Uniform steady state of rest or uniform motion under no external forces; nl9 n2, T, and c0 are all uniform.

b. Steady state at rest under external forces derived from potentials Wl9 ^F2 (1-3) "> in this case T and c 0 are uniform, but not % and w2, which are given by

% = (%V- m i T l / f c r , n2 = (n2\e-^^T (2)

where (wô, (w2) 0 a r e constants (the values of nx and n2 at the points, if any,

where y ¥ 1 or T 2 is zero).

c. A gas in uniform rotation or screw-motion subject to no external force (or to forces whose potentials are constant along the circles or spirals traced out by any point fixed relative to the gas). In this case T must be uniform, but not c0, which is given by

(where co denotes the angular velocity and (c 0 ) 0 the constant velocity at the origin, and wxr denotes the vector product of and #*), nor nl9 n2l which are given by

» i - ( » i ) o T W l T b / * T (3)

where T 0 = — â)2d29 and d denotes the distance of the point r from the

axis of the rotation or screw. Thus the density distribution is the same as if the gas were at rest in a field of (centrifugal) force with this potential.

See also Chapters 10 and 11.

S = — kH0 + constant

(i)

296 KINETIC THEORY OF GASES §2 .2

where U1+ denotes the mean value of any component of Cl9 e.g., Ul9

averaged over those molecules 1 for which this component is positive. Also i hT -

t / ^ r ^ i r ^ y C ^ - , u1v1=v1w1=w1u1 = o (2) so that

(Pxx)i = (Pw)i = (Pzz)i = Pi= k n i T

(Pxy)l = (Pvz)l = (Pzx)l = 0

Thus the stress distribution is hydrostatic, that is, normal and equal across all planes through r.

The " root-mean-square " speed V c x2 is given by

V ( 3 A r / W l ) = 1.086C; (4)

Also \rn~C = \m^ = UT (5)

thus the mean random kinetic energy is the same for both kinds of molecule; this is a particular case of a very general theorem of equipartition of energy.

2.3. The equation of state for a perfect gas. The equation p = knT may be written, in reference to a mass M of the gas, occupying a volume V9

so that n = M\mV, in the form

pV=(kMlm)T (1)

This is the equation of state for a perfect gas, that is, the equation connecting p, Vy and T in the equilibrium state; it includes Boyle's law (pV = constant at any given temperature), and Charles' law (V proportional to T at a given pressure). These laws, however, refer to the thermodynamic temperature, and it is necessary to show that in equilibrium (for which alone the thermodynamic temperature is defined) this is the same as the, kinetic theory temperature T. This can be achieved by considering Carnot's cycle for a perfect gas.

The equation p = knT then confirms, for a perfect gas, Avogadro's hypothesis that all gases in equilibrium, at the same p and T, contain the same number NA of molecules in unit volume. For 1 cc at N.T.P. (0° C and

2.2. Mean values when / is Max well ian. In this case the following mean values (§1.6) of various functions of C are readily found.


1 atmosphere pressure namely, 1.013 X 106 dynes/cm2) this number NA is 2.687 X 10 1 9 ; it is known as Avogadro's number (or sometimes, less appropriately, as Loschmidt's number. See §1.10).

2.4. Specific heats. The symbols cv and Cv(= WcP) denote the specific heat at constant pressure, per gram (cp) or per mole (C^); see § 1.13. That is, cv is the heat required to raise 1 gram (containing \\m molecules) by 1° C ( = dT), at constant pressure. It includes, besides the heat cv

required to increase the mean molecular energy of the molecules, the mechanical energy (pdVjJ in heat units) required to increase the volume by dV, against the constant pressure p> according to Charles' law. The equation of state (§2.3) gives, for M == 1 and dT = 1° C,

pdVIJ=klJm (1)

Thus cp = cv + -j^ (2)

C,= Cv + j£=Cv + !?£ = Cv + R (3)

or Cv — Cv = R (the same for all perfect gases); R is here expressed in heat units (see § 1.11).

The ratio cjcv or CJCV is denoted by y ; by § 1.13,

Thus y = f for molecules with no rotatory or internal communicable energy, for which 1 ; for the diatomic molecules for which (at ordinary temperatures) s = J|, y = 1.4.

2.5. Equation of state for an imperfect gas. The simplest generalization of the equation of state p = knT or pV = (kMjm)T for a perfect gas is

p + ap*=knT(l+bpX) (1) or, less accurately,

(/> + « / > * ) ( i ~ ftp) = fair (2)

which is van der Waals' equation. These equations allow for both the intermolecular forces and the finite size of molecules. For a simple gas in which the molecules are rigid, elastic, attracting spheres of diameter cr, exerting a force F(r) at distance r,


2NU + N12 _ 2N22 + N12 - ° ' (2)

The collision intervals, rl9 r 2 , or mean time interval between collisions for a molecule of kind 1 or 2, are

'f1=llv1, r2=l/v2 (3)

2 1 / 2 W i a i 2 + ( 1 ^ ) 1 / 2 ^ 2 (4)

The mean free paths lx, / 2 , between collisions for a molecule of kind 1 or 2, are

k =

If the gas is simple, / = 0.707/77wa2 (5)

The mean free path /i(Cx) of a molecule of kind 1, moving with random speed Cx is, for various values of 'Cx\Cly as follows :

cjc±: 0 0.25 0.50 1 2 3 4 5 6 Ji(Ci)/ ' i : 0 0.344 0.641 1.026 1.288 1.355 1.380 1.392 1.399 1.414

The probability that a molecule 1 with a particular speed C x shall travel a distance at least equal to / is e - ^ i ( C i ) .

The probability that a molecule 1 moving with any speed should describe a free path at least equal to lis approximately e~1,ml1^.

The fraction of the N12 collisions per unit volume and unit time, between unlike molecules whose collisional energy (namely, their combined trans-

and to a first approximation X = l + f p * (4)

2.6. The free path, collision frequency, collision interval, and collision energy (perfect gas). For a mixed gas composed of rigid, elastic, spherical molecules of diameters aly cr2, ( f ° r kinds 1 and 2), the following results are obtained by use of Maxwell's velocity distribution function (§2 .1).

The number of collisions Nu, AT22, JV12 per unit volume per unit time, between like molecules (1 or 2) or unlike molecules, are given by

where cr12 = + cr2). The collision frequencies, vx, v2, or average number of collisions for a mole

cule 1 or 2 per unit time, are given by

§ 3 . 1 KINETIC THEORY OF GASES 299

latory kinetic energy relative to their mass center, before collision) is x0kT9 is (1 + x*)e-x»\

The mean value of the collisional energy is 2kT.

3. Nonuniform Gas

3.1 . The second approximation t o / . Maxwell's velocity distribution is not correct when T is nonuniform, nor when c0 and the concentration ratio nx\n2 vary from point to point, except for his special states (§§2.1a and2. lb) . Even so, however, it is a good first approximation to / , if n is 10 1 3 or more and if the gradients of T and c0 do not exceed l°/cm and 1 s e c - 1 , respectively. Also, for these conditions, the second approximation to / gives reasonably accurate expressions for the stress distribution, thermal conduction, and diffusion.

The second approximation has the form

/ i = / i ( 0 ) ( i + * i < 1 ) ) a )

w h e r e / 1( 0 ) denotes the Maxwellian expression (2.1), and

4>i( 1 ) = - A ( C , ) C i • grad In T - D^CJC • rf12 - B^C^C^) (2)

where

d12 - grad n 1 0 + ^ mx - m2)f- ^ (tniF1 - m2F2) (3)

and Al9 Dv Bly are scalar functions of the random speed Cl9 determinable from Boltzmann's equation (§ 1.16) and involving molecular interactions. In d12 the vector / denotes the acceleration of the gas at r, namely Dc0IDt9

where D/Dt denotes the hydrodynamic " mobile operator "

d u0d v0d w0d dt dx ^ dy i". dz

f is given by

(5)

300 KINETIC THEORY OF GASES §3 .2

3.2. The stress distribution. From this expression for fv and the corresponding one f o r / 2 , the components of the stress distribution are found to be

7 ™ 2 / 23u0 3v0 dw0\ / dv0 , dwA p ^ k n T - ^ - ^ - - ^ - ^ , P ^ ^ + ^ ) , e t c (1)

where [x is expressible in terms of integrals involving the functions Bx and B2 and depends on the mode of molecular interaction.

These expressions for the stress agree with those of the formal theory of viscous stress, so that [JL is identified with the coefficient of viscosity. In laminar motion, such that u0 = w0 = 0, v0 = azy where a is a constant ( = ô/&r)>

Pxx = Pyy = Pzz = knT, pxy = pxz = 0, pyz = — flU (2)

which is the simplest formal definition of f i .

3.3. Diffusion. In general Cx and C 2 are unequal in a nonuniform mixed gas; that is, the two sets of molecules diffuse through each other. The rate of diffusion is expressible by Cx — C 2 , which is independent of any uniform motion of the axes of reference. It is convenient to consider the flux of either type of molecule relative to axes moving with the mean speed c (rather than c0), in which case

n.C, = - n 2 C 2 = ^ (Cj - Q ) ( 1 )

In this case the expression for Cx — C 2 derived from fx and / 2 gives

n1C1 = — nD12d12 — nDT grad In T (2) where Z) 1 2 and DT are expressible in terms of integrals involving the functions Ax and Z) r and depend on the law of molecular interaction.

This expression shows that diffusion is set up by any of the four following causes : (a) a concentration gradient, (b) acceleration of the gas, (c) unequal external forces on the molecules 1, 2, and (d) a temperature gradient. Diffusion resulting from this last cause is called thermal diffusion. When T is uniform and there is no external force and no acceleration, p and n are uniform (see §§3.1-3.5), and (2) reduces to

nxCY = — nD12 grad n10 = — D12 grad nx (3) This is the formal definition of D12 as the coefficient of diffusion. This coefficient also governs accelerative and forced diffusion, due to the causes ( b ) a n d ( c ) .


Thermal diffusion is governed by a different factor Z>T, called the coefficient of thermal diffusion; the ratio DT/D12 is called the thermal diffusion ratio, and denoted by kT: it has n10n2Q as a factor, and therefore varies greatly with the concentration ratio. The thermal diffusion factor a, defined by kTjn10n20, is much less dependent on nx\n2. Thus

n C ^ - D ^ (4) where

= n grad n10 + m1 - m2)f npp

(m^! — m2F2) + nkT grad In T p

If T is uniform and / zero, and Fx and F2 are derived from potentials y ¥ 1

and *F2, diffusion proceeds in the sense which decreases d12, and if in the limit diffusion ceases, the condition d12 = 0 leads to

g r a d i n g - g r a d ^ ^ ( 5 )

W2 /2i in accordance with Eq. (3) of §2.1.

3 .4. Thermal conduction. Heat conduction in a diffusing mixed gas proves to be expressible most simply by considering the flux relative to axes moving with the molecular mean speed c instead of c0 (when there is no diffusion c and c0 are identical). In this case the expressions for fx and f2 lead to the equation

q = — A grad T — npocD12d12 (1)

where A depends on the mode of molecular interaction, through the functions A-L and Dlt and is identified with the coefficient of thermal conduction in accordance with the formal definition for a nondiffusing gas. The second term in q is proportional to the rate of diffusion; it is an effect inverse to thermal diffusion, and is called the diffusion thermo-effect.

4. The Gas Coefficients for Particular Molecular Models

4.1. Models a to d. The simplest molecular models are :

a. Smooth rigid elastic spheres of diameters al9 a2; let

*12 = Uai + a2)

This is the model already considered in 2.6. For such molecules E = ^mC2,


b. Smooth rigid elastic attracting spheres. This (Sutherland's) model has already been considered in §2.5. The mutual force is F(r) at distance r between two molecules.

c. Point centers of repulsive force, given at distance r by KT~V ; K is called the force constant and v the force index. Suffixes 11, 22, or 12 are to be added to K and v according as the interacting molecules are both mx, both m2, • or unlike.

d. Point centers of repulsive and attractive force, expressed by Kr~v — Krr~v\ where K and v refer to the attraction. Alternatively the interaction may be expressed by the formula 4€[((r7r)v_1 ~~(G'lr)v _ 1 ] ^ o r t n e mutual potential energy of the two molecules; a is the distance at which the force changes sign, and € the potential energy due to either the attractive or repulsive field at this mutual distance. Suffixes 11, 22, or 12 must be added to K, V, K, v .

4.2. Viscosity /x and thermal conductivity A for a simple gas

Model a:

A = 2.522fjicv

Model b (Sutherland's). In this case the expressions for p, and A are those for Model a, divided by 1 + S/T. The term S, known as Sutherland's constant, is proportional to the potential energy of attraction of the molecules at contact. The term S/T is only the first in a series of descending powers of T, and when S/ T is not small, the neglect of the latter terms impairs this interpretation of S, as found empirically from the variation of ft or A with T, as the potential energy.

Model c:

_ A r / ^ I V / 2 (2kTlK)s m^T1^ 8 L \ 77 | A2(v)V(4~s) a

u 2 A r 1 , 3 ( y ~ 5 ) 2 where * = — r and Cv = 1 + ^ _ 1 ^ _ n 3 ) + ...

and A2(v) is a function of v here given for several values of v.

v: 2 3 5 7 9 11 15 21 25 °o

s: 2 1 0.5 0.33 0.25 0.2 0.14 0.1 0.08 0

A2(v): 0.436 0.357 0.332 0.319 0.309 0.307 0.306 0.333

Ax(v): 0.422 0.385 0.382 0.383 0.393 0.5


Also A = fjjLCv

W h e r e ^ T ( i + 4 ( v - ( i H i i r - i 3 - ) + - - - K

and/declines from 2.522 for v = .» (equivalent to Model a) to 2.5 for v = 5.

Model d. In this case, if the attractive field is weak, the expressions for ]i and A are those for Model c, divided by 1 + S/Ta

9 where a = (v — v')j(v — 1), and £ is a function of K, K ' , V, I>'.

In terms of e and or, an alternative expression for JLC is

5 /kmT\112 (£[) * ~~ 16(<7') 2 \ 77 j X € ;

if T is large, the repulsive part of the field is dominant, and x is proportional to (kTje)s. If T is small, the attractive field is dominant, and x is proportional to (&T/e)s', where s' = 2/(v' - 1).

The values v= 13, v = 7 are found to be specially appropriate for many gases. A graph of log 1 0 x is given f ° r these values and also for v = 9, v — 5 in the second item of the bibliography.

The preceding formulas all refer to spherically symmetrical models, for which y = -jj-; this applies to monatomic molecules and also to some diatomic molecules (e.g., H 2 ) at very low temperatures. For other values of y, Eucken has given the following empirical formula for / , which is fairly satisfactory.

/ = l ( 9 y - 5 ) .

4.3. The first approx imat ion to D 1 2 . This is independent of the concentration ratio nY\n2. Its values for particular molecular models are

Model a: 3 /kT(m1 + 0*2)\1/2

a Timr)1^ 8«a 1 2

2 \ 2,7Tm1m2 J nv122

where m = m1m2J(m1 + ra2); rri is called the " reduced " mass of a pair of unlike molecules.

Model b : 3_(kryi2 (2kT/K12y

8n\ mi J At(v)T(3 - s) a wc 1 2 *(m 1 ) 1 / 2 w

where ^ ( y ) has values as in the preceding tablet


The addition of an attracting force to Model a adds a Sutherland factor 1/(1 + S12IT) to the D12 expression for this model.

The correction to the foregoing first approximations to D12 may amount to a few per cent, and it introduce's a small dependence of D12 upon njn2.

4.4. Thermal diffusion. The formula for even the first approximation to hj< or oc is very complicated, and this approximation is less close to the true value than in the case of jit, A, and D12. The correction (which is additive) may be as great as 23 per cent for Model a, when m1jm2 is very large. For other models it is less. And for Model b, when v = 5, kT is zero.

The end values of a, for n10 = 1 and n10 = 0, will be denoted by oc± and oc2. Values of oc for 0 < n1Q < 1 are generally of similar magnitude, though they do not necessarily lie between ocx and oc2. For molecules of very nearly equal mass, and with very different values of cr, it is possible for OLX and oc2

to be of different sign, in which case oc is always very small (e.g., 0.1). Here only the first approximation to oc1 will be given for Model a (that

for oc2 is obtained by changing suffix 2 to 1 except in cr1 2). It is

2 3 / 2 m 1 (w 03 /w 2 ) 1 / 2 ( a 1 / ( 7 1 2 ) 2 — m2\5m2 — Im^

(2/5)K/a1 2)2(2m0/m2)i/2 [ i )

= + (5/23/ 2 ) ( c 7 1 2 / c r 1 ) 2 ( m 2 > 0 ) 1 / 2 ( 7 m 1 - I5m2) Yhm^ + Idmjn^ + 30m2

2

Unless o-j and o2 differ greatly, ocx and oc2 are positive if mx > m2. If there is a steady temperature gradient in a gas mixture, and no external forces or acceleration, diffusion will tend to set up a steady state in which

log n12 = —• a log T + constant

(treating oc as independent of n12 and T, which is usually approximately correct). If a is positive, and the suffix 1 refers to the heavier molecules (m1 > m2), the proportion of gas 1 is enriched in the colder and reduced in the hotter regions, relative to gas 2. Thus the heavier molecules are usually more numerous in the cooler region.

5. Electrical Conductivity in a Neutral Ionized Gas with or without a Magnetic Field

5 .1 . Definitions and symbols. The gas is supposed to consist of neutral particles, ions (positive and negative) and electrons. These constituents are distinguished by suffixes 1, 2, ... attached to the symbols for their properties; mass m\ number density w; charge e\ diameter cr; etc. The suffix 1 will refer to the neutral molecules, so that ex = 0.


The electric intensity is denoted by E, and the magnetic by / / , in the direction of the unit vector h. In the presence of a uniform magnetic field alone, a charge e8 will spiral round the lines of force with angular velocity cos

given by

OJs = esH/ms

The current intensity is denoted by j .

5.2. The diameters crs. The diameter cr1 for the neutral molecules, and also the " joint diameter " or effective distance between centers, crl s, at a collision between a neutral and charged particle (s ^ 1), are a moderate multiple of 10""8 cm (the factor being 1 to 5, diminishing slightly as T increases). The joint diameter aei for a collision between an electron and an ion is of order 10~5 Zi (300/T), where Zt is the number of unit charges on the ion. For a collision between two ions, it is similarly about l0~ 5 Z 4 Z / (300/T) , where Zi and Z are the charge numbers of the two ions.

Owing to the large magnitude of the effective diameters for collisions between charged particles (due to the slow inverse-square decrease of electrostatic force with increasing distance), the collision frequencies of the charged particles are little affected by the presence of neutral molecules unless these are in large excess, e.g., nx more than a hundredfold as great as the number density of all the charged particles, as in a slightly ionized gas (such as the E and F layers of the ionosphere).

5.3. Slightly ionized gas. When n1jn8s^\) is large, the different kinds of charged particles each contribute independently to the current and the electric conductivity, their mutual collisions being negligible. If E is perpendicular to H,

j=KE + K'hxE (1)

where K + iK' = £ n ^ ^ T

An electric field perpendicular to H hence produces a current in its own direction, for which the conductivity is K, and also a transverse current perpendicular both to E and H, for which the conductivity is K'.

If E is along H, there is no transverse current, and K is increased to 2 nses

2rlslms.


5.4. Strongly ionized gas. Consider a gas in which nx is zero or negligible, and the charged particles are electrons, and ions of two kinds, with very different masses. Let suffixes e, i> and j refer to the electrons and the heavier and lighter ions, and take mjnij, m^rrii to be negligible. The electronic contribution to the conductivity is

Ke + iKe' = t y , , . r- (1) ^ e ( l + Tiefae + êîe)

independent of the mutual ionic collisions.

The ion conductivity is similarly given by

ninj(eimj--ejm^ri,

where co^ = + m^\m^ni

(2)

Bibliography

1. BOLTZMANN , L., Vorlesungen uber die Gas Theorie (2 vols.), J . A. Barth, Leipzig, 1910. This pioneer work well shows the nascent ideas of the kinetic theory in their first detailed mathematical development.

2. CHAPMAN , S. and COWLING , T. G., Mathematical Theory of Non- Uniform Gasesy

2d ed., Cambridge University Press, London, 1953. This severely mathematical work gives the classical kinetic theory in detail, but sets out and explains the formulas in a way designed for the use of the physicist and chemist.

3. JEANS, J . H., Kinetic Theory of Gases, Cambridge University Press, London, 1925. This book in its earlier editions stresses the difficulties later met by the quantum theory, and has interesting passages on the statistical mechanical aspects of the kinetic theory. (Dover reprint)

4. KENNARD, E . H., Kinetic Theory of Gases, McGraw-Hill Book Company, New York, Inc., 1938. This book is written for the physicist and does not contain the full mathematical theory.

5. LOEB , L. B., Kinetic Theory of Gases, 2d ed., McGraw-Hill Book Company, Inc., New York, 1934. See remarks on the Kennard book above.

6. MAXWELL, J . C , Scientific Papers (2 vols.), Cambridge University Press, London, 1890. See remarks on the Boltzmann book above.

Chapter 13

ELECTROMAGNETIC THEORY B Y N A T H A N I E L H . F R A N K

Professor of Physics, Massachusetts Institute of Technology

AND W I L L I A M T O B O C M A N Research Assistant, Department of Physics, Massachusetts Institute of Technology

1. Definitions and Fundamental Laws

1.1. P r i m a r y definitions. The study of mechanics is founded on three basic concepts—space, time, and mass. In the theory of electro-magnetism, the additional concept of charge is introduced. Quantity of electric charge is denoted by q, electric current by i, and electric current density by / . The mutual interactions of charges and currents are described in terms of fields of electric intensity E and magnetic induction B. The force on an element of charge caused by all other charges is represented as the interaction of the element of charge with the field (E and B) in its vicinity, the sources of the field being all the other charges. These fields are defined b y

mks Gaussian

f = E+VxB, f = E + r « * (1) dq dq c

where dF is force experienced by an element dq of charge, V is the velocity of the element of charge, and c is the velocity of light in vacuum.

The electromagnetic field in a given region of space depends on the kind of matter occupying the region as well as on the distribution of charge giving rise to the field. This phenomenon can be described in nonconducting, nonferromagnetic media by writing the field vectors as products of two quantitities :

mks

B = ^H = n0KmH,

Gaussian

( 2 ) e KE

B = [iH =. KmH (3)

307

308 ELECTROMAGNETIC THEORY § 1 . 2

where D/e0 and p,JH are what the electric field E and the magnetic induction B would be in vacuum, and the KS are quantities characteristic of the medium. In isotropic media the KS are scalars. D is the electric displacement; H is the magnetic field intensity; K£ is the dielectric constant; KM is the relative permeability; e 0 and /x0 are universal constants

mks Gaussian

e0 = 8.854 X 1 0 - * 2 ^ , e 0 = 1 0 meter 0

ii0 = 47r X l O - 7 ^ ! , u 0 = l r o meter' r o

In terms of the above, the following quantities are defined mks Gaussian

polarization: P = D — e0E, P = ]-(D — e0E) (4) 4 7 7

B 1 / B \ magnetization : M= Hy M = | H \ (5)

magnetic susceptibility .

' =JM . = W ( 6 )

| H | ' j |

electric susceptibility :

x =\ll_ y = . i £ L (7)

L 2 . Conductors . A conductor is a material within which charge is free to move. Metals and electrolytes are conductors; the flow of charge within a metal or electrolyte is governed by Ohm's law :

J = oE (1)

where a (conductivity) is a constant characteristic of the medium, and p = 1/cr is called the resistivity. The resistance of a conductor of length / and constant cross section A is defined

Clearly it is impossible to maintain a static potential difference between two points of a conductor without flow of charge, so that under electrostatic conditions a conductor is an equipotential. This implies that the tangential component of E must vanish at the surface of a conductor for static situations.

§ 1.3 ELECTROMAGNETIC THEORY 309

« • (B2 = 0, n • (Bt -B1) = 0 (1)

« X ( # 2 = /., nx (H2 -Hi) = 477

TJ' (2)

nx (E2 = 0, nx (E2 0 (3)

n • (D2 - J > i ) = os, n • (D2 47T(XS (4) where n is a unit vector normal to the boundary and pointing from region 1 into region 2, as is surface charge density and Js is the surface current density.

1.3. Ferromagnetic materials. The magnetic induction B within a ferromagnetic medium is not a linear function of the magnetic intensity H. The magnetic induction depends not only on the value of the magnetic intensity but also on the previous history of the medium, so that it is possible to have a finite value for B when H = 0. In such a case the body has a residual magnetization Mx = -B//A0 (in mks units) and acts as a source of magnetic field.

1.4. Fundamental laws. Electromagnetic theory can be based on the definitions above and the following three fundamental laws :

conservation of charge: mks Gaussian

X J ~ dt' ^ J ~ dt ( 1 )

Faraday's law :

Ampere's law :

V x / / = / + f , V x * = ^+JL.f (3) ct c c ct

Taking the divergence of both sides of Eq. (2) and (3) and using Eq. (1) there follow Gauss's law:

mks Gaussian V - D = p, V - Z ) = 4ITP (4)

and V • B = 0, V • B = 0 (5) Equations (2), (3), (4), and (5) are called Maxwell's equations. 1.5. Boundary conditions. Application of Maxwell's equations to

infinitesimal contours and volumes at the boundary between two regions in space yield the following boundary conditions for electromagnetic fields.

mks Gaussian

310 ELECTROMAGNETIC THEORY § 1.6

*v-»£-5('-"+*l)-£ Gaussian

9 ~ c* dt2 c dt\ c dt) <= ' mks

(4)

Gaussian (5)

1.7. Bound charge. Combining Eqs. (4) of § 1.1 and (4) of § 1.4 gives for isotropic media

mks Gaussian

V-E= — ( / o - V - P ) , V « £ = — ( p - V - P ) (1) This result has led to the following nomenclature

—V • P = bound charge density p • n = bound surface charge density

where n is the unit outward normal to surface.

1.6. Vector and scalar potentials. From (5) of § 1.4 it follows that

B = V X Ay A = vector potential) (1)

and from (2) of §1.4 that mks ^ Gaussian ^ ^

E = — Vco TTT", E = — V<p - • — ^4, (<p = scalar potential) (2)

These definitions leave V • A (the gauge) undetermined, and the potential cp contains an arbitrary additive constant.

From the fact that the fields must always be finite in regions devoid of charge or current, it follows that the potentials must be continuous functions within such regions.

Using the definition V 2 F = V V - F - V x V x V, (3)

the differential equations relating the potential functions to their sources follow from Eqs. (2), (3) of §1.4, and (1), (2) of §1 .6 ; for homogeneous, isotropic media these equations are

mks \

§ 1.8 ELECTROMAGNETIC THEORY 31 j

^ 4TT\V r Arris r ^ ( r ) • dS (5)

where cp is the potential at a point inside volume V> which is bounded by surface S. Taking S at infinity, where we assume cp vanishes, reduces Eq. (5) to

mks Gaussian

For a point charge, i.e.,

1.8. Amperian currents. Combining Eqs. (5) of § 1.1 and (3) of § 1.4 for a nonferromagnetic isotropic medium gives

mks Gaussian V x B = [i0(J -f- V x 'M)y V x B = 4<T7fjL0(J+ V x M) (1)

when dDjdt == 0.

This result provides the justification for the following nomenclature :

V x M = amperian current density

MX n = amperian surface current density

where n is the outward normal to the surface.

2. Electrostatics

2.1. Fundamental laws. For electrostatic situations, Maxwell's equations for the electric field reduce to

mks Gaussian V x £ = 0, V x £ = 0 (1)

V • D = />, V • D = 4TTP (2)

Since E is now irrotational, it can be derived from a scalar potential :

E=-Vcp ( 3 )

Combining this result with Eqs. (2) of § 1.1 and (4) of § 1.4, there follows Poisson's equation :

mks Gaussian

W = - f v V = - ^ (4)

Integration of Poisson's equation by the Green function method gives

312 ELECTROMAGNETIC THEORY §2.2

equation (6) becomes mks

9 > = w which is called Coulomb's law.

Gaussian 9

<P = er

(7)

2*2. Fields of some simple charge distributions. Equations (7) of § 2.1 and (2) of § 2.1 can be used to give the fields due to various simple charge distributions. Let £ = 1 for mks units, £ = 4TT for Gaussian units.

a. Line of charge (r = linear charge density):

9 InR

b. Sphere of charge (q = total charge; a = radius of the sphere) :

c. Ring of charge Fig. 1 (q = total charge):

9 = 2/ ^—j P n ( c o s « ) P « ( c o s 0), (r > d or 0 ^ a)

(1)

(2)

( r < r f o r M a )

(3)

where P n is the Legendre polynomial.

2.3. Electric multipoles; the double layer. A point charge is regarded as a monopole (2° pole). A 2 n + 1 pole is defined in terms of a 2n pole in the following manner. Let

l"n. = 1

be a vector at the origin. At the tip of ln

place the charge singularity characterized by 2n pole of moment pn; at the back end of ln place a 2n pole of moment —p n \ let ln - > 0 and pn —> oo in such a way as to keep the product lnpn constant.

§2.3 ELECTROMAGNETIC THEORY 313

|>2 = 2/ipj = 2/iô/o (quadrupole)]

where £ = 1 in mks units, £ = 47r in Gaussian units. The potential at point

r = xux + yuy + zuz

of an element of charge dq = pdV located at point

when j r | > | | is

where ^ = 4 ^ I ^ R ? >

(»i • V) B (rt • V ) ^ ...(rr V) 2 ( f l • V ) ^ )

( 5 )

( 6 )

by Taylor's theorem. The subscripts simply serve to count the number of factors.

A double layer is a surface distribution of dipoles. Let r be the dipole moment per unit area; then the potential due to the double layer is

9 = -^-Jrdii (7)

where dQ is the element of solid angle intercepted by an element of area of the double layer.

The resulting charge singularity is called a 2n+1 pole of moment

Pn+l = (» + WnPn (1)

The potential resulting from such a singularity is

= «« ' V«„-! • V«„_2 •... • « 0 • V ( 1 J (2)

Accordingly,

90 = 4 ^ - ^ ' l>o = ? (monopole)] (3)

<Pl = ~ i e P l U o ' V ( T ) ' I > I = M (dipole)] (4)


4TT€ n r q H Z " + 1 (r > rx)

(1)

b. Dielectric sphere of inductive capacity ex in the field of a point charge q> both immersed in a medium of inductive capacity e :

2.4. Electrostatic boundary value problems. The electrostatic field potential in a given region must satisfy the following conditions :

a. cp must be a solution of Poisson's equation, Eq. (4) of §2.1.

b. The change in (eVcp * n) must be equal to the surface charge density across any boundary.

c. cp must be constant in a conductor.

d. cp is continuous everywhere (except across a double layer).

e. cp vanishes at least as fast as 1/r at infinity if all sources are within a finite region. These conditions determine the field uniquely.

2.5. Solutions of simple electrostatic boundary value problems. Let

£ — 1 for the mks system of units,

£ = 4TT for the Gaussian system of units.

a. Conducting sphere in the field of a point charge, center at the origin, where

rx — radius of sphere q1 — net charge on sphere charge == q at z = Z > rl9 y — 0, x = 0

r2 = Vr^+~Z* - 2rZ cos 6

$ = potential* at (r,9,<p)

§2 .6 ELECTROMAGNETIC THEORY 315

c. Conducting sphere of radius rx in a uniform electric field:

 r i )

(D (3)

(r < 4776^!

where the external field has the potential

<D0 = —E0r cos 9

d. Dielectric sphere of radius rt in the uniform electric field :

<S>0 = — E0r cos 6

<D = - E0r cos 6 + ^ = £ - rfE0 ^ L , (r > rj «i + 2e

E0r cos # (r < rx)

(4)

Let 2^ be the field within the sphere, and let Px be the polarization of the dielectric sphere (example d). The depolarizing factor L, which is defined by the equation

LP1 = E«-E1 (5)

is then

for a sphere

L = 3e 0 /c(/ci — 1) (6)

2.6. The m e t h o d of images . For the case of a regularly shaped conductor or dielectric in the field of a simple distribution of charge, it is often possible to construct a potential distribution which satisfies the boundary conditions outside the conductor or dielectric by replacing the dielectric or conductor ^ by a distribution of charges located °' inside of its boundaries. Since the field so constructed satisfies Poisson's equation as well as the boundary conditions, it must represent the actual field outside the conductor. Such fictitious charge distributions are called image charges.

/ / / / / / / / / / / / /

FIGURE 2


FIGURE 3

Image charges of charge q at

...

Image charges of charge — q at

3. Point charge outside a conducting sphere:

q' = image charge = — OR = — - (Fig. 4) , OQ OQ

5

FIGURE 4

Examples: 1. Point charge outside an infinite grounded plane conductor, Fig. 2,

p. 315 (q = image charge = —q). 2. Point charge outside a pair of intersecting grounded plane conductors,

Fig. 3; (n = integer):

5. Point charge near a plane boundary between two semi-infinite dielectrics, Fig. 6; (Let k == €2l€i):

Region I

a

Region II

FIGURE 6

The field in region II is given by replacing region I with charge q" :

n* 2 k a

The field in region I is given by replacing region II with charge q' :


2.7. Capaci tors . A capacitor is a system consisting of two insulated conductors bearing equal and opposite charge. Let A 9 be the difference of potential between the two conductors, and let q be the magnitude of the charge on each. Then the capacitance of the system is defined to be

C = I <•> The capacitance of a single conductor is defined as the limit of the capacitance of a system of two such conductors as their separation becomes infinite. The work required to charge a capacitor is

w=±ci±<pY = Y i (2)

The capacitances of some simple capacitors may be calculated by the formulas shown below, where £ = 1 for the mks system, and £ = 4TT for the Gaussian system.

a. Parallel plates:

A — area of the plates

d = the separation of the plates

C = | j , (A>d*) (3)

b. Concentric spheres:

rx = radius of inner sphere

r 2 = radius of outer sphere

C=UVrT-Vr2) (4)

c. Concentric circular cylinders :

rY = inner radius

r2 = outer radius

L = length

C = IIin?7^1-' ( L > r > - r J (5)

d. Parallel circular cylinders :

Same symbols as in example c. D — separation of centers; positive


when the cylinders are external, negative when one cylinder is inside the other

2 Y \ L • 2 r i r 2 J ) I

L>D and [r^-r^)

e. Two circular cylinders of equal radii:

4TT€ L

C ' 4

Cylinder and an infinite plane :

h = separation of center of cylinder and plane

h

(6)

C = ~ ' ^ ( c o s h - ^ y \ (L>D) (7)

r 4TT<E L_ c o s h - 1 -i r

1 (L>h) (8)

g. Circular disk of radius r : C = 4 ^ . 2 r ( 9 )

2.8. Normal stress on a conductor. Consider an element of surface dA of a conductor. Let the charge density on this element be ex. Then there will be an outward force dF normal to dA :

mks Gaussian

dF=(^-^)jadA)> M=(?£p)(crdA)

The normal stress on such a surface is

mks Gaussian dF o-2 dF _ W 2

~dA ~ 2 ? ~dA~~~2^ (1)

2.9. Energy density of the electrostatic field. The energy of a static system of charges is just the sum over all the charge of the potential energy of each element of charge due to the field of all the other charge. This energy is related to the field by the equation

mks Gaussian

W =

The energy density of the electric field is therefore taken to be

mks Gaussian ED ED • .

« = — , u = - ^ - (2)


4TT)V r I

- A ! ^ X - ^ + < " - > > < ^ ) + « - V ( I ) H (5)

where A is the vector potential at a point inside a volume V bounded by surface 5, and n is the unit outward normal to the surface. Letting S approach infinity where we assume A must vanish reduces Eq. (5) to

A

mks Gaussian dv A a r wdv

The potential of the current element ids = Jdv is then

mks Gaussian

dA = -£-ids, dA=^-ids ( 7 ) Airr cr v 7

* STRATTON , J. A . , Electromagnetic Theory, McGraw-Hill Book Co., Inc., New York, 1941, p. 250.

3. Magnetostatics

3.1. Fundamental laws. For magnetostatic situations Maxwell's equations for the magnetic field reduce to

mks Gaussian

V x # = / , V x # = — (1) c

V - B = 0, V • £ = 0 (2)

Since B is divergenceless it may be derived from a vector potential

B = Vx A (3)

Then for the case of a nonferromagnetic, isotropic, homogeneous medium, Eq. (1) becomes

mks Gaussian

V M = - u 7 , V*A = - ^ J (4) where

V2A = VV • A - V x V x A and we choose V • A = 0. Equation (4) can be integrated by the vector analogue of the Green's function method * to give

§ 3 . 2 ELECTROMAGNETIC THEORY 321

so that mks Gaussian

dB dB = ^ /jut dsx r

(8) c r3

Equation (8) is called the Biot-Savart law.

3.2. Fields of some simple current distributions. From Eqs. (6) and (8) of § 3.1, we can deduce the fields caused by various simple current distributions. Let £ = 1 in mks units, £ = 4TT in Gaussian units; let £ = 1 in mks units, £ = l/c in Gaussian units.

a. Infinite straight wire :

H = ^ ^ R (».)

b. On the axis of a circular current loop (Fig. 7 ) : iR2

(2) 2(x2 + R2)*!'

X

FIGURE 7

c. On the axis of a solenoid (Fig. 8 ) :

ff=£y(cos tfi + cos d2) (3)

where n = number of turns per unit length.

FIGURE 8


3.3. Scalar potential for magnetostatics; the magnetic dipole. For regions where / = 0, H is irrotational and may therefore be derived from a scalar potential function i/j.

J 5 T = — ( 1 )

The potential due to a loop of current is then mks Gaussian i i£l , iQ, „x

<A = - T - > <A = (2) where f is the current in the loop, and O is the solid angle determined by the loop and the point of observation. The sign of Q is determined as follows. Choose an origin at the center of the loop (Fig. 9). Let r be the vector from the origin to the point of observation; let a be the vector from the origin to the current element ids. Then Q has the same sign as a X ids • r.

If the area A of the loop approaches zero and the current i in the loop, approaches infinity in such a way as to keep iA constant,

mks Gaussian . iA cos 6 iA «• r iA

where n is the unit normal to A. ax ids a X ids I

(3)


V(l/r) — • f P X Jdv, 7T J

(4)

J where p is the radius vector from the origin to the current element Jdv = ids. T o find the vector potential of a magnetic dipole :

- S « | ( - < ) + « [ ^ ' ( T ) ] I = - £<«) („ • V)V ( | ) = - 1 x [« x V ( 1 ) ]

where m = t4n, and = 1 in mks units; £ | = 47r /c in Gaussian units. It is seen that the vector potential of a magnetic dipole is

mks Gaussian

(5)

3.4. Magnet ic mult ip les . Let |£ = 1 for mks units; |£ = ATT/C for Gaussian units. The vector potential dA at the point (x,y,z) due to the current element Jdv = ids at the point (#i,J>i,#i) when

r = Vx2 + y* + # 2 > rx = V * i 2 + Ji 2 + V is

W ( * - * i ) 2 + (y - yi)2 + (* - „~0

By analogy with the expression for the potential due to the electric dipole, Eq. (4) of § 2.3, iA is called the magnetic dipole moment.

The preceding expressions are readily generalized to describe an arbitrary steady-state current distribution (which can always be regarded as consisting of a collection of current loops).

mks Gaussian N

V(V') x

324 ELECTROMAGNETIC THEORY § 3 . 4

where

dAn

and

4TT (ri • V)„ (»-! • V ) ^ ... • V ) 2 ( r i • V ) x j ) ] d v

If all the current is confined to the interior of a sphere of radius JR, then for r > R

(2) uo

= 2 A*#(-I y

477 -J / [(r 1-V) n (r 1-V) B_1 . . .(r 1 .V)1(L|

If the current distribution is stationary, the first term, A0, vanishes since in that case the current distribution can be analyzed into closed current loops and for the kth. such loop

±ds = î<bdrt = 0 r Airr T 1

The contribution of the kxh current loop to A± is

A . = _ ^ | R , . V ( L ) * ,

fi^H f (rxX drjx V(l/r) 477

477

J1 ( 3 )

where Ak is the area of the M i loop and nk is the unit normal to the loop. Thus by Eq. (5) of §3.3,

is the vector potential of the magnetic dipole moment of the current distribution. Accordingly, An is identified with the vector potential of the magnetic 2n pole moment of the distribution. Clearly, the magnetic mono-pole does not exist.


B0sm I/J , (r < a) i

(1)

A, = £ \ - % ^ + B0r sin f j (r > a) Ax — Ay — 0

where £ = 1 for mks units; £ = 1/c for Gaussian units.

3.5. Magnetostatic boundary-value p r o b l e m s . In regions where J =0 the scalar magnetic potential must satisfy the following conditions :

1. V2«/f = 0 at all points not on boundaries between two regions of different jLt.

2. I/J is finite and continuous everywhere.

3. The normal derivatives of ifj across a surface are related by

4. iff vanishes at least as fast as 1/r2 as r becomes infinite, if all sources are confined to a finite region.

If / ^z£ 0 one must use the vectors B and H, which satisfy the following conditions :

1. V X H = J (mks); V X H = Air\c J (Gaussian).

2. B = pbQ(H + M) (mks); B = pu0(H + M ) / 4TT (Gaussian), where M

is the magnetization.

3. Across any boundary

A(#i X H) = JS (mks); A(/i X H) = 4TTJCJS (Gaussian)

A ( « • B) = 0 (mks); A(n • B) = 0 (Gaussian)

4. If sources are confined to a finite region, B vanishes at least as fast as 1/r2 as r becomes infinite.

These conditions determine the magnetic field uniquely. In the absence of sources the solution IF of a magnetostatic boundary

value problem becomes identical with the solution cp of an electrostatic boundary value problem for the case of no charge when e is replaced by JJL.

As an example of a solution of a magnetostatic boundary value problem involving current, consider the field that results when an infinitely long straight wire of radius a, permeability / X j , and with a current i is embedded in a medium of permeability /x2, that contains an external field B0 directed transverse to the axis of the wire.


L = b U (4)

2. Solenoid, where radius = ay length = /, number of turns per unit length = w.

L = TTa*im2[s/P + a2 - a] U (5) 3. Parallel wires, where radii of the wires — a, c\ separation of the

centers — b\ permeability of the wires = /x •= permeability of the external medium.

4. Coaxial cable, where inner radius = a, outer radius = b.

L — 2[JL In (—\ = self-inductance per unit length (7)

The demagnetization factor Lx is denned in a manner analogous to that for the depolarization factor, Eq. (5) of § 2.5,

L±M = H0-H±

where M is the magnetization of the body in question, H± the field within the body, and HQ the uniform external field.

3.6. Inductance. The quantity JL? • dA = for a circuit is called the total flux linkage and is generally proportional to the current giving rise to B. The self-inductance of a circuit is defined by the expression

(i)

where i is the current in the circuit. Similarly the mutual inductance between circuit 1 and circuit 2 is defined by

M l 2 = m ^ ( 2 ) h

where ( « 0 ) 1 2 is the total flux linkage of circuit 1 caused by current i2 in circuit 2. It can be shown that

M 1 2 = M 2 1 (3)

The inductances of some simple circuits are listed below, where = 1 for mks units; = Airjc for Gaussian units.

1. Circular loop, where radius of loop = b, radius of wire = a, \L = magnetic permeability of the wire.

§3 . 7 ELECTROMAGNETIC THEORY 327

5. Two closely wound coils on a ring of permeability /x, where radius of the ring = a, radius of the cross section of the ring ==. b, and n and m are the respective numbers of turns in the two coils.

M^ fjLnm[a-(a2 -~b2yi*]tt ( 8 )

6. Two circular loops (Fig. 10).

FIGURE 1 0 M = $^7TjjLa sin oc sin /?

%nn\i) ( y ) " P ^ C 0 S a ) p « 0 ( c o s f t p « ° ( c o s y )

3.7. Magnetostatic energy density. The work required to establish current i in a circuit of self-inductance L is

W=±-Li2 (1)

If we regard this energy as being stored in the magnetic field produced by the current• t, the energy density of the magnetic field is

mks Gaussian dW 1 dW 1

This type of energy is the electromagnetic analogue of mechanical kinetic energy.


4. Electric Circuits

4.1. The quasi-stationary approximation. The analysis of circuits is usually based on the assumption that the electric and magnetic fields at a given moment are essentially the same as would be produced by the charge and current distribution at that moment if these distributions were fixed in time. This is called the quasi-stationary approximation and is valid where the dimensions of the circuit are small compared to ejeo, where c is the velocity of light and w is the angular frequency of current variation.

4.2. Voltage and impedance. The difference of potential between two points of a circuit is called the difference of voltage AV between these points. The impedance Z between two points of a circuit is the ratio of voltage drop to current between these points :

It follows that the impedance of two circuit elements connected in series is

Z = Z 1 + Z 2 (2)

and the impedance of two circuit elements connected in parallel is

Z = l / Z 1 + 1 / Z 2 ( 3 )

4.3. Resistors and capacitors in series and parallel connection. Resistors:

series : R — R1 + R2 + ^ 3 + = total resistance (1)

1 = 1 + 1 R R± R2

parallel: -^- = -^- + -=L+ . . . = (total resistance)-1 (2)

Capacitors:

parallel: C = C1+ C 2 + C 3 + ... = total capacitance (3)

series : ^ = ~ + ^ - + ... = (total capacitance) - 1 (4)

4.4. Kirchhoff's rules. Direct current circuit analysis is based on KirchhofFs rules :

a. At any branch point of a circuit, the sum of the currents entering equals the sum of the currents leaving the junction.

b. The sum of the voltage drops around any closed loop of the network equals zero.

ELECTROMAGNETIC THEORY 329

4.5. Alternating current circuits. Consider a circuit having a source of potential causing a voltage rise E between two points of the circuit. Let R be the resistance of the circuit, C be the capacitance of the circuit, and let L be its self-inductance. Then by Faraday's law,

JErdr = - j d V = - L § = ( ± + i R - E )

or E = iR + L~ + ^ jidt \ (1)

E _ d2 R_ dq 1 T ~ d t 2 < 1 + L' dt + CLq

The solution of this differential equation can be expressed as the sum of two parts

a = <lt + <ls = transient solution + steady-state solution

There are three possibilities for qt:

a. R2 > 4L/C (overdamping)

(2)

= ' f t - A g o ) e P i t + _(/>*?«-»o) eVlt (3)

where ^ = _ _ + _ ^ _ _

(4)

* 1 / m 4L

and <?0 and % are the initial values of charge and current, respectively.

b. R2 = 4L/C (critical damping)

fc= [<7o + (*o + ^ ? o ) ' ] * - * < / 2 L

c. i ? 2 < 4L/C (underdamping)

*« = \ ( o T ^ + - ) s i n w i < + ?o « » 6)^1 r - * ^ (5) I y ZL/COj O ^ / J

U 1 / l ^ where o>, = , \ /1 j y -

The steady-state solution is

„ _ E0 sin (wt + ifj) q* L[(u0

2-a>2)2 + (a>0ojlQ)2YI2 W


where j = V ~ 1.

For the case where

E = E0e^t

the impedance for the steady state is

The instantaneous power delivered from the voltage source is

The average power is

E2

p = iE = i*Z = ^ (9)

/>AV = ^ = ^COS0, (10) where E0 and i0 are the amplitudes of the respective periodic functions, and

n .coL — 1/coC .... g ^ . t a n - 1

i ?/ (11)

For a given circuit the current is greatest at the circular frequency to0>

(12) T JL/

in which case Z = R.

5. Electromagnetic Radiation

5.1. Poynting 's t h e o r e m . From Maxwell's equations and the vector identity

V(ExH) = HVxE-EVxH

it can be shown that mks

V - ( £ x H)=-~ . - i - E D + BH)-EJ

V - ( £ x # ) = ~ y y ( « - D + BH)-EJ

where E = E0 cos cot; Q = co0L/R; co0 = l / V Z C ,

0 , (ifJB = £0e*««) (7)


F=jy(pE + Jx B)dv, F=jv(pE + J^*))dv

It is possible to reduce this equation to the following form.

(i)

F -- - ^dv + l jjs(X-dAi + Y-dAj + Z- dAk) (2)

where £ = 1 in mks units, £ = 1/477 in Gaussian units, and

$[e(E/ - Ef - E*) +p(Hx* - H* - tf,»)] = £(eExEv + pHxHy) r«

l(eExEy + pHxHy)

- - iS,») + KH,' - H/ - H*)] = ^ 2 2

(3)

For the case of stationary fields and charge distributions, the first term on the right of Eq. (3) vanishes, and it becomes apparent that TTJ is only the equilibrium electromagnetic stress tensor.

On the other hand, if the surface S is taken to be at infinity, the second term on the right of Eq. (3) vanishes, and

mks Gaussian _ r BS , dP . €fi r dS , dP

This is called Poynting's theorem and is interpreted by identifying

mks Gaussian

S = Ex H, S = ^ ( £ x H), (Poynting's vector) (2)

with the flux density of electromagnetic energy. In addition it can be shown that

mks Gaussian

Ve/x \ 2 2 7 Any/€ii- \ 2 2 /

5.2. Electromagnetic stress and momentum. Suppose we have a charge distribution p(x,y,z>t) in a homogeneous, isotropic medium. Then the total force of electromagnetic origin on the charge contained in volume V is

mks Gaussian


mks Gaussian

where G = €/JL $ Sdv, G = ^$Sdv (4)

and P is the total mechanical momentum. But since the boundaries of the system have been taken at infinity, the

system must be closed. Hence its momentum must be conserved. One is thus led to identify G with the electromagnetic momentum. Accordingly, the electromagnetic momentum density is

mks Gaussian

g = € j uS, g = %S (5)

5 3 . The Hertz vector; electromagnetic waves. A completely general description of an electromagnetic field is provided by the specification of the four scalar functions that comprise the vector and scalar potentials A and cp. Very often one deals with fields that have symmetry properties rendering them susceptible to a simpler description involving less than four scalar functions. The basis for such a simplified description is provided by requiring the scalar potential or its counterpart to be some specific function of the vector potential. T o see how this may best be done, perform a gauge transformation with an arbitrary function / yielding a new vector potential

A = Aoi<i-Vf

and a new arbitrary scalar potential

<p = cpoia + -^f

for which Eqs. (4) and (5) of § 1.6 assume the form

V - ^€Â-P + €V + £ V < p ) - 0 , (Gaussian)

where P = $Jdt (3)

so that / = dP/dt, and p == - V • P by Eq. (1) of § 1.2.

(1)

(2)


d2

V X V X n + e/x - T - g l l - P + eVcp = 0 (mks) dt2

€/X A V A ll 7 -

If the gauge V X V X n + —2U - ArrP + eV<p' = 0 (Gaussian)

J (7)

cp = - V • n/e (8)

is chosen, Eq. (7) reduces to a form particularly convenient when II is expressed in rectangular coordinates.

mks Gaussian

vm-,e^n = - P , V* - ^ • | r n = - 4 . P (9)

where W — V v — „ , „

dt ' c dt ( (10)

When V • P — —p = 0, Eqs. (1) and (2) of § 5.3 assume a simple form if / is chosen so that A=0 V • A = 0

> (4)

V A - ^ A — * VA-p&A—**! \ Thus the vector potential is useful for the simplified description of fields in the absence of charge.

Clearly Eqs. (1) and (2) of § 5.3 are fulfilled if it is required that

V x V x fAdt+€[iÂ-iJ&+€iiV (5)

V x V x fAdt+^Â-^P + ^V<p = 0 (Gaussian) J In this way it is possible to reduce the four equations (4) and (5) of § 1.6 to three equations. Equation (5) has a simpler form when expressed in terms of the Hertz vector II.

mks Gaussian

Jl = —iAdt9 U = —iAdt (6) /x J /x J

In terms of the Hertz vector, Eq. (5) becomes


In a region where P vanishes, Maxwell's equations are symmetrical in E and H and in B and D, so that in such regions there is a second possible field represented by the Hertz vector 1^.

mks Gaussian

B = - V V • n , + v * - n X ) = £ ( - W . n , + f ^ n t )

where .

V ^ - ^ n ^ o , v ^ - ^ ^ - n ^ o ( 1 2 )

Inside an isotropic, homogeneous conductor free from external sources, we again are able to have two distinct types of solutions, each of which can be expressed in terms of a Hertz vector,

mks

E = E l + Et = - ? ^ * * + lXx±iil

H = H1 + H2 = Vx V x n , + V x 8tN2 + Y V x n 2

V 2 N 2 ~~ W U * ~ I 1 1 * = 0

Gaussian

£ = £ 1 + £ 2 = ^ V x 5 - - ' x V x n 2

H ^ ^ + H ^ — Vx V x n r + — V x | - n 9 + — V x n 2

C C DT " €C

( 1 3 )

(14)

( 1 5 )

( 1 6 )

Since the most general separable time dependence can be represented as a sum or integral over simple harmonic terms, let

n = n c e - ^ * > V = I ) ( 1 7 )


OJ c

(21)

For (crjcje) << 1 this equation represents a wave traveling with phase velocity

mks Gaussian

where c = velocity of propagation of light in free space and n =z\/f<£KM is called the index of refraction.

5.4. Plane waves . T o find the expression for a plane wave traveling along the x axis in a source-free region, let

n 0 = oi + oj -\-Tl(x)k

Then Eqs. (13), (14), and (18) of 5.3 have the solution

mks Gaussian

K2 K2

E2 = — e±jxxk, E2 = — e±jxxk

K2

H1 — — K2e±j*xk, H± = — — e±jxxk

(i)

Then Eqs. (14) and (16) become the vector wave equations

mks Gaussian

V ^ + ^ c o ^ l + ^ I l o ^ O , V ^ + ^ ^ l + ^ y ^ O (18)

vm0 + k2[\+^n0==o, v 2 n 0 + ^ 2 ( i + ^ j n 0 ^ o (19)

V 2 n 0 + K 2 n 0 = o, V 2 n 0 + K 2 n 0 - o (20)

where k = V/ê OJ k = \/Ji€ -


AC yJ€ + (ja/coy c2 K c y e + (j4rra/w)

where Z is called the wave impedance.

(4)

5.5. Cylindrical waves . Let R, i/j, z be the parameters of a circular cylindrical coordinate system. T o find the representation for cylindrical waves in a source-free region let the Hertz vector be

n 0 = OI + oj +U(Ribz)k

Then Eq. (18) of §5.13,

V 2 H 0 + K 2 n 0 = 0 becomes V 2 n + K2U = 0

which has the solutions n (n) = einy, e^ZJ^K2 — h2 R) (1)

where A is a separation constant and Zn is a solution of the nth order Bessel equation. From Eqs. (13) and (14) of § 5.3, we see that this gives two solutions:

Transverse magnetic

mks Gaussian .7 an<w> _ jh an<»>

3R ' R 4TT dR

E = — nUn) E = —nl\n)

Ez=(K2-h2)Il^ E. = V £ ) U I N ) \ ( 2 )

ZJ K * T T f « » rr K2CnIJ^ fJLtoR 4TTfJLtoR

an<n> _ JK2C dii(n)

V ~~ fJUO ' 3R ' v ~ 4 7 7 y I c o ' ~ W ~

= 0, J5T, == 0

The general solution is thus seen to satisfy the relations

E = EY + E2, H = H1 + H2

E'H=E1H2 + E2H1 = Q (2)

ixE = ZH (3) where

mks Gaussian


Er ER

/aw«II ( M )

Er R ' ER

AttR

EV

. 6n<»> EV

jfuo 8 n ( n >

~~~c*~ ~8R

EZ = 0, EZ = 0

HR = j h 8R ' Hr

jh 811 <"> c dR

= - - £ * n < « » , HV

H„ = (/c 2 -A2)n<«>, H, _ ( K

2 - A 2 ) n < " >

? P)

J

5.6. Spherical waves . Let r, 0, 9 be the parameters of a spherical coordinate system. T o find the representation for spherical electromagnetic waves in a source-free region, take the Hertz vector to be

n o = n o M > < p K + oue + °u<p

Then Eq. (8) of 5.3 becomes

6 8 r " + r 2 s i n 0 a ^ S m f / 80 f2 sin2 0 8<p2

r 80 r 86\dr ) I U ;

8<p^ = 0

r sin 0 Sep r sin 6 dep

T o satisfy the last two equations we need require only that the arbitrary scalar potential cp0 be equal to — (1/e) (dn o/dr). With this substitution, the first equation reduces to the form

V 2 ( 5 » ) + K 2 ( 5 O ) = O (2)

The general solution of this equation is

r Vkt > (3) (fi = 0 , l , 2 , 3 , . . . ; « = 0 , .1 ,2 , . . . ) >

Transverse electric

mks Gaussian

fir drdV

1 < fir sin 8 drdtp'

„ 1 a2(n0) ' r r r 1 H * = ^ v z ^ w H EE = ATTEQ of mks system

£ ' = ^ r £ v ^ = 4 ^ r o f mks system ^ <5>

Hr = 0, i/ r — (47rjc)Hr of mks system

# e = JC*J—n ' i r ^ # 0 = (4TT/C)H9 of mks system r sin u dcp '

H<p ' H v = Arrjc)!!^ of mks system

where / is an integer.

where Zn+aj2) is a solution of the (n + ^) order Bessel equation, and Pnm is

the associated Legendre polynomial. From Eq. (13) of §5.3, we see how these solutions may be used to give

two kinds of spherical waves. Tranverse electric

mks Gaussian Er — 0, Er = Er of mks system

Ef> = ^ n ' ~i-^> Ed = EQ of mks system v r sin 8 dcp

Ey =^ — y • Ey = Ey of mks system

Hr = — | ^~^-°+ a c 2 I I 0 j Hr = c i / r of mks system

= /(/- + -i)n0

jLtr2 '

1 ^2If He=— -T^, He = cHe of mks system


5.7. Radiation of electromagnetic waves; the oscillating dipole. T o find the expressions relating electromagnetic radiation in an isotropic medium to its sources we must solve the inhomogeneous vector wave equation for the Hertz vector.

where £ = 1 for mks units; £ — 4TT for Gaussian units. This is equivalent to the three inhomogeneous scalar wave equations for

the rectangular components of n and P.

V2n* _ i ' = ~ *p" ( i = 1>2,3) ( 2 )

A solution of this equation is

(3)

^ 4tt Jsl r 1 \ rj ^ vr \ dt J ndS

where (x\y\z') is the point of observation in volume V bounded by surface S> r = V(x — x'f + (y — y'f + (z — z')\ f* = f(x,yyzyt — rjv), and n is the unit outward normal to surface element dS.

In an unbounded medium this becomes just

(4) all

space

If the time dependence of P is simple harmonic P = erJ»*P0(x,y,z) (5)

then

N = J J J P» XYTRLV d x d y d z ^

7 7 all r

space

An important example is the radiation field of an oscillating electric dipole of moment p =poe~jtot along the z axis.

/> = e~M J = — en*"* j zV •

Taking the surface infinitely far from the origin

P = e-w $PQzdv


And since Jv = Jx = 0, we must have PQy — P0x = 0 so

kp = $Pdv = ^P0e^wtdv

Substituting this into Eq. (6) of 5.1, we get

Applying Eq. (10) of § 5.3 to the above Hertz vector yields the following field for the electric dipole of moment p = pQe~ia)t along the z axis.

J>o*3

4-776 •2;

= ^ j e - « » « - K D s i n 0 47TC act* (fcr)2 ( k t ) 3 (7)

=/_ + _±_ KT (/cr)2

where £ — 1 for mks units, £ = 477 for Gaussian units; £ == 1 for mks units, £ = \\c for Gaussian units.

5.8. Huygen 's pr inc ip le . If ^ is a solution of the scalar equation

i , ^ ( 1 )

then xji satisfies the equation

(2)

where r = (x — x')i + (j> —y')j + (# —V)A: = r • ru

f* = f(x>y,*,t — rlv)

n = unit outward normal to element of area dS

This is called Huygen's principle. Suppose Q = 0 in F and \fj is the spherical wave.

g-jmit-rjv)

where ' i

r1 = (x- x1)i + (y - yi)j + ( z - zx)k = rx T^- = r l W l

(3)

§5.9

Then

ELECTROMAGNETIC THEORY

' 5 ATTYYX

For light waves it is generally the case that 1/r, l /r x << co/e;, so that

4W»T 1 t> (« • w — « i • »)rf<S

341

(4)

(5)

5.9. Electromagnetic waves at boundaries in dielectric media. Given a plane electromagnetic wave

E = E0e-j(a>t-k*r) = (Ey + Exz)e-j0it-k*r)

whose propagation vector k lies in the xyz plane impinging on the boundary between two dielectrics which coincides with the xyy plane, let n — n^ for z < 0, n = n2 for z > 0, and let k • « > 0 where w is a unit vector parallel to the z axis (Fig. 11).

Let

[ n2

r\

— u

FIGURE 11

, k* u l == COS""1 -r-TT

be the angle of k with respect to « . Then there can only result a refracted wave (&') a reflected wave (k ") or both. The angles these waves make with the normal to the surface are related by

nx sin i =. n2 sin r, (Snell's law) (2)


/ 2 sin r cos i .r X \ = 8 i n ( f + r ) l f ^ = ^ j ( 5 )

2 sin r cos £ \ sin(* + r) cos(i

where Ey is the component of E0 normal to the plane of incidence (xyz plane) and Exz is the component of EQ tangent to the plane of incidence.

When the angle of incidence equals

iB = t a n - 1 — , (Brewster's angle) (7)

xz

so that the reflected light is polarized. When the angle of incidence exceeds

s i n - ^ (8) n

total reflection occurs, and the refracted wave vanishes. For further optical formulas, see Chapters 16 and 17.

5.10* Propagation of electromagnetic radiation in wave guides. Let the z axis be the axis of the wave guide. Take the Hertz vector to be

n 0 = oi + o / + n o * (1)

Then II 0 must be a solution of the scalar wave equation

V 2 n 0 + Acn0 = 0 (2)

Since there are no obstacles to propagation along the z axis, one can separate out the z dependence to get

II 0 = e^kizu(xyy) (3)

The amplitudes of the refracted and reflected waves are related to that of the incident beam by FresnePs equations :

^ = M l t a n / - M 2 t a n r / s i n ^ - , ) \ hy tan z + tan r \ sin(* + r) r J _ 7

E"xz _ [LX sin / cos i — /z 2 sin r cos r / tan(z — r) _ \ Exz ^ sin i cos i + /u,2 sin r cos r \ tan(z + r) Pi )

E' y 2JU,2 sin r cos t Z? /Xj sin j cos r + JJL2 sin r cos i

E'xz 2/x2 cos i sin r E x z JJL2 sin r cos r + /xx sin i cos *

• + ( £ ) • ] . - (4)

(5)

(6)

§ 5 JO ELECTROMAGNETIC THEORY 343

where u is a solution of the equation " a2 d2

3x^Jr"dy2

The expressions for the field vectors are derived from the Hertz vector by means of Eqs. (10) and (11) of § 5.3.

Transverse electric (TE)

+ j

Transverse magnetic ( T M )

where f — 1 for mks units, | = \\c for Gaussian units. It is seen that

EH = 0, Ht = ^ ^ (7)

where the subscript t identifies the transverse components and

Z O ^ ^ A / A (for the T M mode) (8) Ag y €

Z0'=^JJL9 (for the T E mode) (9) A0 \ e

The function u(x,y) depends on the shape of the cross section of the wave guide and on the material of its walls. If we suppose the walls to be perfect conductors,

^tangent — -^normal — 0

at the boundaries so that for T M waves

,x-r \ U ^ I at the boundaries (10) (Vtt)tan = 0 ) v ' and for T E waves

(Vw)normai = 0 at the boundaries (11)


(4) all

space

A ( w , t ) = & JJJ - r ^ d X l d y l d Z l ( 5 ) all

space

where r is the magnitude of the vector

r = (x1- x)i + yx - y)j + (zx - z)k

The expressions above are called the retarded potentials. When the source of the field is a point charge of magnitude q, velocity

r = (d/dt)r, and acceleration r = (d2/dt2)r; then the expressions for the retarded potentials can be integrated to give the Lienard and Wiechert potentials.

A = & J . J (7)

5.11. The retarded potentials; the Lienard-Wiechert potentials; the self-force of electric charge. If the gauge

mks Gaussian

V A = - V % V A — * . $ ( I ,

is chosen, Eqs. (4) and (5) of § 1.6 take the form of inhomogeneous wave equations :

mks Gaussian

V2A 1 3 2 A T X72A 1 d * A 4 7 T J n \

TO 1 #V P V72 1 d \ 4 7 7

where v — l /V/^ f ° r t n e m k s s v s t e m > ^ = CW~\**- f ° r t n e Gaussian system. We obtain the general solution of these equations by adding a special solution of the inhomogeneous wave equation to the general solution of the corresponding homogeneous wave equation. The special solution of the above equations which relates the electromagnetic field to its sources is


The fields resulting from these potentials are

"(I -r*jc2) (r + rr/c) 47T€ (r + r 9 rjc)3

r (r + rf/c X r) c2 (r + r • *V 0

(JEx r). t-^r/c

(8)

(9)

The expressions above can be used to compute the interaction of a distribution of charge with its own field. For this purpose we postulate a rigid distribution of charge such that all elements of charge dq have the same velocity r and acceleration f. It is necessary to assume that the average linear dimension of the charge distribution r0 is so small and the motion of the charge varies so slowly that the change of acceleration in the period of time it takes a light wave to pass the charge distribution is small compared to the acceleration itself.

I — r\<\r\ I c I 1 1

On the basis of this assumption the charge distribution is found to exert the force F on itself :

•F=F0 + F1 + Ft+...

where dqdq'

4TT€ (10)

'^jjrdqdq'* q'rr0 (11)

where F0 is the term representing the inertia of the field of the charge, Fj is independent of the form of the charge distribution and accounts for the damping resulting from the radiation of energy.

Perhaps the most important point is that this chapter has been prepared with the view of giving the reader an insight into the coherent structure of the whole subject of electromagnetic theory, rather than presenting him with an unrelated set of fundamental formulas in the field.

346 ELECTROMAGNETIC THEORY

T A B L E 1

CONVERSION TABLE FOR SYMBOLS

To convert an expression from its form in the mks system to its proper form in the Gaussian system, and vice versa, make the following substitutions

mks Gaussian mks Gaussian E - > E J —• / B - > B/c D — K DjAn A —• H —• HCJATT s - > 5

€ ejArr n -> n / 4 7 T

fxAnjc2 c - > G/4TT

P p L X/c

T A B L E 2

CONVERSION TABLE FOR UNITS

mks units Gaussian units

time, t . 1 sec = 1 sec length, / l m = 102 cm mass, m 1 kg = 103 gm force, F 1 newton = 105 dynes energy, w 1 joule •= 107 ergs power, p . . . . . . . . . .. . 1 watt = 107 ergs/sec charge, q 1 coulomb = 3 X 109 statcoulombs electric field, E 1 v/m = 1/3 X 10~4 statvolt/cm electric displacement, D 1 coulomb/m = An x 3 X 105 statcoulombs/cm2

potential, <p 1 v = 1/3 X 10 " 2 statvolt capacitance, C 1 farad = 9 X 10 1 1 cm current, 1 amp = 3 X 109 statamperes resistance, R . 1 ohm = 1/9 X 1 0 - 1 1 statohm magnetic field, H 1 amp/m = An X 1 0 - 3 oersted magnetic induction, B 1 weber/m2 = 104 gausses magnetic flux, BdS. . . . . . . 1 weber = 108 maxwells inductance, L . . . 1 henry = 109 abhenrys


T A B L E 3

T H E FUNDAMENTAL CONSTANTS

permitivity in vacuum, e 0

permeability in vacuum, (JL0

velocity of light, c . . . . charge of the electron, e. . mass of the electron, m0. .

mks

8.854 X 1 0 - 1 2 farad/m 4TT x 1 0 - 7 henry/m 2.998 X 108 m/sec 1.601 X 1 0 " 1 9 coulomb 9.108 X 10 " 3 1 kg

Gaussian

1 1 2.998 x 10 1 0 cm/sec 4.803 X 10~ 1 0 statcoulomb 9.108 x 10~ 2 8 g

T A B L E 4

COORDINATES AND SYMBOLS

Unless indicated otherwise, the following systems of coordinates and symbols were used in this chapter.

Rectangular coordinates (Fig. 12), r1 = ixi—jyx—zji :

FIGURE 12

348 ELECTROMAGNETIC THEORY

Auz

/ i ^ X ^

FIGURE 13

Spherical coordinates (Fig. 14) :

FIGURE 14

J A / . 1 > y \ = 1 in mks units ( = 4TT in Gaussian units, ^

j = 1 in mks units / = \\c in Gaussian units

Circular cylindrical coordinates (Fig. 13) :


Bibliography

1. ABRAHAM , M. and BECKER, R., The Classical Theory of Electricity and Magnetism, Blackie & Son, Ltd., London, 1932. A standard intermediate text.

2. BECKER, R., Theorie der Elektrizitat, Band II, Elektronentheorie, B. G. Teubner, Berlin, 1933. The classical theory of the electron, containing much material not to be found elsewhere.

3. BORN , M., Optik, Julius Springer, Berlin, 1933. An advanced comprehensive book based on electromagnetic theory.

4. FRANK , N. H., Introduction to Electricity and Optics, McGraw-Hill Book Company, Inc., New York, 1950.

5. JEANS, J., The Mathematical Theory of Electricity and Magnetism, Cambridge University Press, London, 1948. A classical treatment of electrostatics and magnetostatics, presented without the use of vector analysis. (Dover reprint)

6. SLATER, J. C , Microwave Electronics, D. Van Nostrand Company, Inc., New York, 1950. Contains material on transmission line theory and circuit analysis, as well as electromagnetic waves.

7. SMYTHE , W . R., Static and Dynamic Electricity, McGraw-Hill Book Company, Inc., New York, 1950. An excellent reference text for methods of solution of advanced problems.

8. STRATTON , J. A., Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, 1941. A valuable treatise on the subject.

9. BLATT , J. M. and WEISSKOPF, Theoretical Nuclear Physics, John Wiley and Sons, New York, 1952. Appendix B provides an excellent discussion of multipole radiation.

Chapter 14

E L E C T R O N I C S

B Y E M O R Y L. C H A F F E E Professor of Applied Physics

Harvard University

1. Electron Ballistics

Since the field of electronics extends over the areas of physics and engineering, many of the basic formulas in electronics dealing with specific fields will be found listed under these fields, such as atomic physics, nuclear theory, particle acceleration, conduction in gases, electron microscope, solid state, microwave spectroscopy, etc. This wide spread of the subject of electronics makes it difficult to select basic formulas not to be found duplicated elsewhere. The formulas chosen are those which are fundamental and of general application in the broad field of electronics.

1.1. Current. A stream of charged particles composed of n particles per unit length of the path, moving with a velocity v and each particle having a charge q constitutes a current / .

I=nqv (1) The current density / is

J=pv (2) where p is the charge density.

1.2. Forces on electrons. An electron having a charge — es located in an electric field Es in esu is acted upon by a force F.

F = -esEs dynes es = 4.803 X 1 0 - 1 0 esu (1)

or F=—emEm dynes em == 1.602 X I O - 2 0 emu (2)

where Em is in emu. An electron having a velocity v cm/sec in a magnetic field of flux density B

in emu is acted upon by a force F.

F = -emv X B dynes em = 1.602 X 10~ 2 0 emu (3)

350

§1 .3 ELECTRONICS 351

1.3. Energy of electron. The loss in potential energy or gain in kinetic energy U of an electron having a charge —em while moving from point xx to point x2 in an electric field Em in emu is

U: - em f*2 Emdx = - em Vx - V2) ergs J *i a )

where Vx and V2 are the potentials in emu at xt and x2, respectively.

The kinetic energy of an electron moving with velocity v cm/sec is

1

1

Vl-(vlcf

3

- 1 ergs

1 . + • T M ( T ) ' + ... ergs (2)

where tn0 = rest mass of the electron = 9.106 X 10~2 8 gram, and c = velocity of light = 2.9978 X 10 1 0 cm/sec.

The transverse mass mt of an electron moving with velocity v is

mt =

The longitudinal mass ml of an electron is

nil = [i-(*A0T/ a

(3)

(4)

1.4. Electron orbit. An electron having a velocity of v cm/sec in a magnetic field of flux density B moves in a stable circular orbit of radius r cm, when F = — mtv2lr (see § 1.2), giving for the cyclotron condition

where m0 = 9.108 X 10~? 8 gram (see § 1.3).

t = W**.r cms i'„, B

(1)

(2)

where V is the electron velocity v expressed in equivalent volts.

The periodic time T of one revolution of an electron in the cyclotron orbit is

352 ELECTRONICS § 2.1

2.2. Cylindrical electrodes. The space-charge-limited current from a cylindrical central emitter of radius a cm to a concentric cylindrical anode of radius r cm, neglecting initial velocity of emission, is

, 2 \~2e~s IV^2 , ,

IV 3'2

= 14.68 X 1 0 - * - ^ - amp

where / is the length of the plate,

2. Space Charge

2. L Infinite parallel planes. When space charge exists between parallel planes d centimeters apart', one being an emitter of electrons having no initial velocity, the space charge limited current is

/ = TT~ 1 • —r2- (esu) J 9TT m d2 v '

(1) 77 ,3/2 « V '

= 2.336 x 1 0 - 6 —d--~ amp/cm2, (Child's law) where Vb is the potential difference between the planes in volts. Relativity change in mass is neglected.

Variation of field, velocity, and space charge with distance x from emitter is

Ex = — (constant) • * l / 3 (2)

vxr= (constant) • x2ls (3)

px — (constant) • x~~2'3 (4) where p is the space-charge density.

When initial velocity of emission is not neglected, the space-charge-limited current is

^"2,336 x mJEfcW ( ' + « 6 ^ ^ 3 + - ) " ^ - * ( 5 )

where Vm is the potential with respect to the emitter of the potential minimum xm cm from the emitter; k being Boltzmann's constant (1.3804 X 10~ 1 6 erg/deg), and Tthe absolute temperature of the emitter.

§ 3.1 ELECTRONICS 353

and

^ = l o « f - f ( , o g f ) ' + ra>(los T)' - 5150 fa i f + •••'*

Variation of field, velocity, and space-charge density with r is

Er = (constant) • ( r £ 2 ) 2 / 3 (2)

*V = (constant) • (rj3 2 ) l / 3 (3)

p r = (constant) • (r 2 jS)" 2 / 3 (4)

3. Emission of Electrons

3.1. Thermionic emission. The saturation current density is

/ = ^ r 2 ( l - r)e-®«IkT€-ElM2lkT amp/cm2 (1)

where A = Airme^h3 = 120 amp/cm2 deg2

T = absolute temperature in degrees C r = reflection coefficient of electrons at the potential barrier of the

surface <E> == electron affinity or potential barrier in esu e = electronic charge in esu k = Boltzmann's constant (1.3804 X 10~1 6 erg/deg) E — electric field at the surface in esu h = Planck's constant (6.625 X 10" 2 7 erg sec)

The factor e- £ l / 2 e 3 / 2 / f c r gives the effect upon emission caused by the electric

field at the surface, i.e., the " Schottky effect.^

3.2. Photoelectric emission

l- mv2 — hv — + Uk ergs (1)

where m = mass of the electron v = maximum velocity of ejected electron in cm/sec h — Planck's constant v = frequency of incident radiation

0 = electron affinity or potential barrier in esu e = charge of electron in esu

Uk = kinetic energy of electron inside metal

* For table of jS2 see LANGMUIR, Phys. Rev., 21, 435 (1923) and LANGMUIR and BLODGETT, Phys. Rev., 22, 347 (1923).

354 ELECTRONICS §4.1

PT7 2

dV 2 == ( JO S*N ^ ^ ) (JO Z ^ C 0 S ^ ) 77

where i(X) is the current in Z,0 caused by unit change in Cvf.

dco (2)

Bibliography

1. Dow, W . G., Fundamentals of Engineering Electronics, John Wiley & Sons, Inc. New York, 1937.

2. FINK , D . F . , Engineering Electronics, McGraw-Hill Book Company, Inc., New York, 1938.

3. HUGHES, A. L. and DUBRIDGE , L. A., Photoelectric Phenomena, McGraw-Hill Book Company, Inc., New York, 1932.

4. MAXFIELD, F . A. and BENEDICT, R . R . , Theory of Gaseous Conduction and Electronics, McGraw-Hill Book Company, Inc., New York, 1941.

5. MI LLMAN , J. and SEELY, S., Electronics, McGraw-Hill Book Company, Inc., New York, 1951.

6. M. I. T. Electrical Engineering Staff, Applied Electronics, John Wiley & Sons, Inc., New York, 1943.

7. SPANGENBERG, K . R . , Vacuum Tubes, McGraw-Hill Book Company, Inc., New York, 1948. Advances in Electronics, edited by L. Marton, Academic Press, New York; Vol. I, 1948; Vol. 2, 1950; Vol. 3, 1951.

4. Fluctuation Effects

4.1. T h e r m a l noise . The mean-square voltage in frequency interval dv across an impedance having a real component of R ohms is

dV „ 2 = 4 x lQ~7RkT dv volts2 sec (1)

dVJ = — x \0'7RkT dw volts2 sec (2) 77

where T is the temperature in degrees Kelvin and k is Boltzmann's constant, (1.3804 x IO" 1 6 erg/deg), and to = 2TTV.

4.2. Shot noise . The mean-square voltage across a pure resistance R ohms in the plate circuit of a diode operating under space current saturation is

F 2 = ^ volts2 (1)

where e is the electronic charge in coulombs, / is the average plate current in amperes, and Cvf is the capacitance between plate and cathode in farads.

If the impedance in the plate circuit is Z in parallel with Cpf, the mean-square voltage in the frequency range 277 dv = dco is

i 2 l

Chapter 15

SOUND AND ACOUSTICS B Y P H I L I P M. M O R S E

Professor of Physics Massachusetts Institute of Technology

1. Sound and Acoustics

The science of acoustics bears a close relationship to may other fields of physics, as covered in this book of fundamental formulas. For example, hydrodynamics provides the basic equations that govern the flow of sound in an acoustic system. There are problems of dynamics and boundary values must be satisfied. Thermodynamics, statistical mechanics, and kinetic theory have a relationship in terms of the microscopic picture of acoustics. Reference to certain of these other chapters, therefore, may assist the student. The following formulas were chosen as being the most fundamental to the physical side of acoustics. The purely engineering phases, of course, properly lie outside the scope of this work, if for no other reason than the fact that many of them are empirical.

1.1. Wave equation, definitions. For a gas or fluid with negligible viscosity, where v(r>t) is the velocity of displacement of the fluid, at r,t, from equilibrium, P 0 is the equilibrium pressure and p(r,i) the deviation from this, the equation determining the velocity potential i/r, for fluctuations of p small compared with P 0 , and the relationship between I/J and other quantities is

where K = pc2 is the compressional modulus of the fluid and Ap is its departure from equilibrium density p. For a gas, vibrating rapidly enough, the departures from equilibrium are adiabatic, every change in pressure p brings about a change in temperature A T from the equilibrium tempera-

c2V2*js = (aty/a**),. C* = (KIP) (1)

v = -grad</r, p = pidifj/dt), Ap = (plc2)(dilj/dt) (2)

355

356 SOUND AND ACOUSTICS § 1.2

ture T. For a gas having ratio of specific heat (cjcv) — y, the congressional modulus and temperature change are

K = p C 2 = Y P 0 > \T = Tlc*)y-\)d4,ldt) = Tlpc*)y-\)p (3)

1.2. Energy, intensity. The energy density of sound in the fluid is

E = + i(c*iP)p*

= ^p(\ grad i/f|2 + c2\difj/dt\2), (energy per unit volume)

The power flow through the fluid, the intensity of sound

S = pv = — p(di/jldt) grad (power per unit area) (2)

When intensity can be measured directly (when both velocity and pressure are measured) the magnitude of S is often given in terms of the decibel scale.

Intensity level — 10 log (intensity in microwatts per sq cm) -f- 100 (3)

An intensity of 10~ 1 0 microwatt per square centimeter is zero intensity level, of 1 watt per square centimeter is 160 intensity level, etc Sound of zero intensity level at 1000 cycles per second frequency is just at the threshold of hearing for the average person. The ear is less sensitive to frequencies either higher or lower than 1000 c. For example, the threshold of hearing is at about 40 decibels for 100 c. and at about 10 decibels for 10,000 c. Above about 20,000 c. and below about 20 c. sound is not heard.

1.3. Plane wave o f sound. The " standard " wave motion is the simple-harmonic, plane wave, of frequency v — (LOJITT) = (kcj2ir), wavelength A == (lirjk) =.cjv) and direction parellel to the propagation vector k.

$ = Aeih-r-»t\ p = peHk-r-a>i)9 v = Um&x(kjk)eik'r-(ot)

Pmax = —iptoA, £ / m ax = (Pirmx/pc) = tkA,

( A T ) m a x = ( T / ^ ) ( r - l ) P m a x

E - (Pmax 2 /2/* 2 ) = ipt7 m ax 2 , S - (P^/lpc) = \pcV^2 = Ec

In most cases intensity is not measured directly; pressure amplitude P m a x is measured. Sound is then measured in terms of sound pressure level, which is adjusted to be equal to the intensity level for a plane wave in air at standard conditions.

Pressure level = 20 log ( P m s / 2 X IO - 4 ) = 20 log P r m s + 74 (2)

where Prms = Pmax /V2 is the root-mean-square pressure amplitude measured in microbars (dynes per square centimeter). Note that for any

§1.4 SOUND AND ACOUSTICS 357

other condition except a plane wave in air at standard conditions, pressure level is not equal to intensity level.

1.4. Acoustical constants for various media. The pertinent sound propagation constants in cgs units for various media are as follows. Air at 760 mm mercury pressure, 20° C temperature; p = 0.00121, c — 34,400, pc = 42, pc2 = K== 1,420,000, y = 1.40, P 0 = 1.013 X 103. Sea water, at 15° C temperature, 31.6 g salt per 1000 g water (standard conditions); p = 1.02338, c = 150,000, pc = 1.53 x 105, pc* = K =, 2.3 X 10 1 0 .

1.5. Vibrations of sound producers; simple oscillator. A simple oscillator has mass M grams, resistance R dyne second per centimeter, and spring compliance C centimeter per dyne. The equation of motion is

M(d2xjdt2) + Rdxjdt) + (x/C) = force (1) The free vibration is

x = Ae~kt sin (curt + cp) (2) where the damping constant k = (R/2M) and the vibration frequency is (a> r /27r) , where ojr

2 = (1/MC) — k2. The Q of the oscillator, the number of cycles for the amplitude of motion to reduce to e~n of its initial value is Q = VMICR2. For forced motion, if the driving force is Fe-i0)\ the displacement and velocity of the oscillator are x — Fj—i(joZ)e~ioit and (dxjdt) = (FjZ)e~i<ot

y respectively, where Z is the mechanical impedance of the oscillator.

Z - - ^ M + P - ( l / ^ C ) - > a > M + P + (l/ya)C), (i = -j) (3)

where M is analogous to electric inductance, and C to electric capacitance in a series RLC circuit. When M or a> is large enough for a>M to predominate, the oscillator is mass-controlled and Z ^jwM\ when C or o> is small enough, the oscillator is stiffness-controlled and Z ~ (l/jwC).

1.6. Flexible string under tension. For mass e gram per centimeter length, tension T dynes, the transverse displacement y(xj) of a point x on the string, from equilibrium, is given by the wave equation (neglecting friction)

c2(32yldx2) = (d2yjdt2\ c2 = (T/e) (1)

where c is wave velocity. For a string clamped at x = 0 and x = /, the possible free vibrations are

00

y(x,t) = ^ An sin (mrx/l) cos [(nirctjl) — i/fn] (2) n=l

The allowed frequencies are vn = (OJJITT) = (mill), the nth being called

358 SOUND AND ACOUSTICS § 1.7

nth harmonic of the fundamental frequency vx = (c/2l). The amplitudes An

and phases <pn are determined by the initial shape y0(x) and velocity v0(x) of the string.

An cos \jjn = (2//) jlQyoix) s m (Trnx/l)dx (3)

^4W sin if;n = Ijmrc) *;0(#) sin (irnx/fydx (4)

For a transverse driving force F = F0ei(Ot dynes per centimeter applied uniformly along the string of length /, the steady-state driven motion of the string is

The mean displacement is

^ ) = ( F 0 ^ 7 £ o > 2 ) [ ( 2 c / a > / ) t a n ( ^ (6)

having resonances at the fundamental o>, and at all odd harmonics co 3 ,a> 5 , . . . .

1.7* Circular membrane under tension. For mass a per unit area and tension T per unit length, the transverse displacement y(rycp,t) from equilibrium of a circular membrane clamped at r == a, for free vibration, is

y(r,<p,t) = 2 [Amn cos (imp) + £ m w sin (mcp)]Jm(7rPmnrla) • )

m,n \ (\)

c o s [ ( 7 r ^ r o n r t / a ) —0 m w J J where the term / w ( # ) is the Bessel function of order m, and j8 m w is the nth root of the equation Jm(7rfi) = 0. The resonance frequencies are vmn = (ajmn/27r) = (fimnc/2a). The first few values of j8 are : j80 1 = 0.7655, j 8 0 2 = 1,7571, j 8 u = 1.2197, j81 2 = 2.2330, Pmn-+n + \m-\ where » > 1 .

For a transverse driving force F = F0e~ia>t dynes per square centimeter, uniform over the membrane, the steady-state displacement, after transients have disappeared, is

and the mean displacement is

^ ) = ( ^ - < » V ) [ A ( W » W ] (3) Resonance occurs at o> = oj0n = (7Tj8 0 n c /a) .

§1 .8 SOUND AND ACOUSTICS 359

1.8. Reflection of plane sound waves, acoustic impedance. If a plane sound wave, of pressure amplitude P i n c , is incident on a plane surface at angle of incidence 6, a reflected wave is produced, with amplitude dependent on the reaction of the surface to the sound pressure. This reaction may be expressed in terms of a ratio z between the pressure at the surface, p at x = 0, to the normal velocity of the air at the surface, vx at x = 0 (in many cases z is independent of 9). The pressure wave is

p Pjn c^^(^cos 6+ysin d-ct) ^

(zjpc) COS 0 — 1 Pin (1)

eik(-xcos 0+2/sin 0—ct) l (zjpc) COS 0 + 1

If St is the incident intensity (Pinc2l2pc), the reflected intensity is

_ I zjpc) cos 0 - 1 I2 N

^ ~ ^ | ( * / p c ) c o s 0 + 1 I

which is equal to S when z is imaginary (surface impedance a pure reactance). When # has a real part, &r is less than St and the surface absorbs some of the incident power.

1.9. Sound transmission through ducts. For a pipe, with axis parallel to the x axis, of cross-sectional area A(x) and length of cross-sectional perimeter L(x), the inner surface of which has acoustic impedance zw(x\ when the logarithmic derivatives (A'/A), (L'/L), and z'jz) are small compared to (1/L), the following wave equation is approximately valid.

For a simple-harmonic wave p = P(x)e~ltot where (to = kc), this reduces to

When A is independent of x but (A/L) is small compared to (c/co) = 27rA, the motion for a wave proceeding in the positive x direction is

P = P m 2 b X e m n x - c t ) — PMGIXE-(LJ2A)(Qe/zVmk(x-€t) (3)

The reciprocal of zW9 Vzw) = g — w> *s t n e acoustic admittance of the duct surface; the wave is attenuated if the conductance g is greater than zero; the effective phase velocity of the wave in the duct is cj[\ + (^/2^4) which is greater thaa c if s is negative (mass reactance), less than c if s is positive (stiffness reactance of wall).

The attenuation of the wave in decibels per centimeter is A3ALpcgjA)y

with L, A, etc., given in centimeters (since peg is dimensionless, if L and A

360 SOUND AND ACOUSTICS §1.10 are given in feet, A3ApcgLjA will be the attenuation per foot of duct). If the tube is closed at x = I by material (or an abrupt change of the duct) of effective impedance (pltvx)x=*i — zh there will be a reflected as well as an incident wave in the region x < /. The pressure and velocity at x,e can be written

p = PQ sinh [ikn(x — I) + i/j]e-i<ot \ v = (nP0lpc) cosh [ikn(x -I) + xfte-™1 \ ^

where i/j = tanh - 1 (nzjpc). The ratio of p to v at x = 0 is then

#o = (pcln)tann [ tanh - 1 (nzjpc) — ikn\ (2) which would be proportional to the load on a diaphragm, driving the air in the tube, placed at x = 0.

1.10. Transmission through long horn. When zw is large enough so that n is nearly unity, but when A varies with x, we have the case of the horn. For the conical horn A = A0[(x/d)-]~ l ] 2 the solution for an outgoing wave is

p = Po[(XA) + = ( i ^ = r T T ^ ) (1)

For an exponential horn S = S0e2xld and the solution is

p = PQe-xld+ik(rx-ct\ zQ = j ^ - l = per — ipccJLod) (co > eld) (2)

where r = V1 — (ofcod)2 and kc - co. If the duct of the previous section is terminated with a conical or exponential horn at x = /, the # 0 of this section is to be inserted for zx in the previous equations.

1.11. Acoustical circuits. A duct with various constrictions, openings, and divisions is analogous to an a-c electric circuit as long as the wavelength of the sound is long compared with the dimensions of the tube. The total flow of air, A(x)vy past any cross section, is analogous to the current, and the pressure is analogous to the voltage. The analogous impedance Za = (zjA) may be computed in terms of the impedance of the analogous electric circuit. A constriction in the tube, consisting of a hole of area A in a baffle plate, is analogous to a series inductance of value

La = (plJA), / . = / + OAVJ (1) where / is the length of the constriction (/ == 0 when the plate is of negligible thickness). A chamber of larger cross section (a tank), of volume V, is analogous to a shunt capacitance of value

Q = ( *7 /* 2 ) (2)

§1.12 SOUND A N D ACOUSTICS 361

A Helmholtz resonator is a tank with a single hole, connecting to the outside, is analogous to a series LC circuit. The resonance frequency is

vr = (1/277VLJCa) = cJIttWAJIJ (3)

A muffler, consisting of a sequence of tanks, connected one to the next by constrictions, is analogous to a low-pass filter network of series inductances alternating with shunt capacitances. Frequencies above the cutoff frequency

VC = (HWLJCa) = (cMVWeV (4) are attenuated, those below vc are transmitted.

1.12. Radiation of sound from a vibrating cylinder. When a long cylinder of radius a oscillates with a velocity U0e~l(Ot transverse to its axis, the sound wave radiated from its surface into an infinite medium is

P - ( = ^ ) < W ) [ f f , » > ( * r ) ]

when a is small compared to (IttjX) = (I Ik). Cylindrical coordinates, r, cf> are used, <j> being the angle between radius r and the plane of vibration of the cylinder. The intensity of the sound several wavelengths from the cylinder, and the total power radiated per unit length of cylinder are

Sr ^™*Pa*U*l4ch) cos*<t>, n ^ ( 7 7 2 o ) 3 p « 4 t / 02 / 4 c 2 ) ( 2 )

The reaction force F of the medium back on the moving cylinder, per unit length, is proportional to its transverse velocity U; the ratio (F/U) is the radiation impedance per unit length of the vibrating cylinder.

Zrad — — ico(7ra2p) + ( ? 7 2 c o 2 / o a 4 / 2 £ 2 ) , (wa << c) (3)

1.13. Radiation from a simple source. Any source of sound of dimensions much smaller than a wavelength of the radiated sound sends out a wave which is almost symmetric spherically if there are no reflecting objects nearby. The source strength Q0 is the amplitude of the total in-and-out flow of fluid from the source, in cubic centimeters per second. The radiated wave, a distance r from the radiator is

p ^ — iaj(pl47Tr)Q0emr-ct\ (kc == o>) (1)

The corresponding intensity and total power radiated are

Sra*(pu,*Q0*l32n*a*), Tl - pu^Q^c) (2)

362 SOUND AND ACOUSTICS §1.14

The radiation impedance back on the sphere, the ratio of the net reaction force to the velocity of the sphere, is

Zrad — — ( f w 3 ) + (rrco^pa6l3c% (coa > c) (2)

1.15. Radiation from a piston in a wall. If a flat-topped piston, of circular cross section with radius a, set in a flat, rigid wall, vibrates parallel to its axis with velocity U0e~ta)\ the radiated pressure and power at a distancer (r a) from the center of the piston, r at angle 8 to the piston axis, and the total radiated power are

gikr—ct)

p ~ —itopU0a2 —-— [J±(ka sin 8)jka sin 8]

-r—>-ia)p(7Ta2UQl27rr)e i k ( r - c t )

S ^ i(p^2C/(,2a4/r2) [Ji(ka sin 0)/*a sin 0] 2 *• puPlbfia*) (mfi- U0f

( l )

n == \(ira2pcU2) [1 - (Ijka)A(2fei)], (ka = coafc = iTrajX)

The radiation impedance, the ratio of reaction force on the piston front to the piston transverse velocity, is

^ r a d = Tra2pc[R0 — iX0]

R ° " 1 - \Ta)U2ka) ~* j 1 (Aa > 1) ^ (2)

v 4 f 1 / 2 j r - / i t \ • o J ( ( 8 ^ / 3 7 7 ) , ( f e l < l )

A n = — sin (2ka cos a) sur a doc -> , 1 '.. ^ .: 0 77 J O V 7 ( (2 /T7&2) , ( & t f > l )

1.16. Scattering of sound from a cylinder. A plane wave of sound, of frequency COJ2TT\ incident at right angles to the axis of a rigid circular cylinder of radius a is in part underflected (continuing as a plane wave) and in part scattered. For wavelengths long compared with the cylinder radius a,

1.14. Radiation from a dipole source. A sphere of radius a oscillating back and forth along the spherical coordinate axis gives rise to dipole radiation when lirr is small compared to the wavelength. If the linear velocity of the sphere is U0e~ta>t

y the radiated pressure, intensity, and power, several wavelengths from the source, are

§1.17 SOUND AND ACOUSTICS 363

the intensity Ss of the wave scattered at angle <p to the direction of the incident wave and the total power scattered per unit length of cylinder are.

Ss ~ ( 7 r c o 3 a V 8 ^ r ) S 0 ( l - 2 c o s ^ ) 2 , Us ^ (37r2co3a4.S0/4c3), (ka<\) (1)

where S0 is the intensity of the incident plane wave. For wavelengths short compared with a, half the scattered wave interferes with the undeflected plane wave, forming the shadow of the cylinder (with diffraction at the edge of the shadow) and the other half is reflected with distribution of intensity

Sr ^(aS0/2r)sin(</>/2), I I r ^ 2aS0, (ka>\) (2)

The net force on the cylinder because of the wave, per unit length of cylinder, is in the direction of the incident wave of magnitude

( ioo(47TWIc)P0e-io>\ (ka<\)

( V%-nac\a> p ^ - ^ / c H a - * * ) * . / ^ ^ a > 1)

where P 0 is the pressure amplitude of the incident plane wave.

L I 7 . Scattering of sound from a sphere. The corresponding scattered intensity and power from a plane wave of frequency (OJ/ITT) = (kcîr) incident on a rigid sphere of radius <z, at distance r and angle & to the incident wave are

Ss (ojWS0j9c*r2) ( 1 - 3 cos# ) 2 , US ^ (16TTO)WI9c*)S0 (1)

for wavelengths long compared with <z, where ka << 1. For very short wavelengths half of the scattered wave produces the shadow and the other half has intensity and power scattered,

Sre*(a*l4t*)S0, N r ^ 7 r a 2 . S 0 , (ka > 1) (2)

The net pressure on the sphere, caused by the incident wave is

Pa ^ (1 + I ika cos &)P0ei(O\ (ka<\) (3)

1.18. Room acoustics. When sound is produced in a room the average intensity eventually comes to a constant value when the power produced is just balanced by the power lost, most of which is lost by absorption at the room walls (for the acoustic frequencies; high-frequency sound is absorbed in the air). If the room walls are sufficiently irregular in shape the steady-state sound is fairly uniformly distributed in direction and position in the room. The absorbing properties of the wall, in this case, may be expressed in terms of the absorption coefficient a, the fraction of the incident intensity which is absorbed and not reflected. This coefficient is related to the acoustic impedance of the wall, as defined previously. If an area As of the wall

364 SOUND AND ACOUSTICS

(or ceiling or floor) has absorption coefficient ocs, the sum (ocÂ^ OL2A2 + ...) over the whole surface of the room is called a, the total absorption of the room. If A S is given in square feet, the units of a are called sabines. The mean intensity level in the room, for steady state, is

Intensity level ^ 10 log (Ufa) + 130 decibels (1)

if II is the power output in watts and a is in sabines. If, after reaching steady state, the sound is turned off, the intensity decays exponentially, on the average; the time required for the intensity to diminish by 60 decibels,

T = (0.049 V\a) seconds (2)

is called the reverberation time (V is the volume of the room in cubic feet). The reverberation time is a useful criterion (not the only one, however) for the acoustical properties of an auditorium. Satisfactory values lie between 0.7 second and 1.5 seconds, the, higher values being more appropriate for rooms used for music, the lower for rooms used primarily for speaking. A reverberation time less than 0.5 second indicates an undue amount of absorption and a corresponding lack of room sonority.

Bibliography

1. BERANEK, L. L., Acoustic Measurements, John Wiley & Sons, Inc., New York, 1949. A good modern text on experimental acoustics.

2. BERGMANN , L., Ultrasonics, John Wiley & Sons, Inc., New York, 1944. A discussion of the newly developing field of very high frequency sound. Contains a good bibliography.

3 . FLETCHER, H., Speech and Hearing, D. Van Nostrand Company, Inc., New York, 1929. An excellent discussion of the physiological aspects of acoustics.

4. KNUDSEN, V. O. and HARRIS , C. M., Acoustical Design in Architecture, John Wiley & Sons, Inc., New York, 1950. A recent review of an important and rapidly developing application of acoustics.

5. MORSE , P. M., Vibration and Sound, 2d ed., McGraw-Hill Book Company, Inc., New York, 1948. A text on the recent developments of the theory of wave motion and its application to acoustics.

6. RANDALL, R . H., Introduction to Acoustics, Addison-Wesley Press, Inc., Cambridge, Mass., 1951. An up-to-date and easily readable introductory text.

7. RAYLEIGH , J. W . S., The Theory of Sound (reprint), Dover Publications, New York, 1945. The classic of the science of acoustics.

INDEX

This comprehensive index covers the two volumes of the book. Volume One contains pages 1 through 364 and Volume Two contains pages 365 through 741.

ABBE'S SINE CONDITION, 376 Abel integral equation, 99 Aberration of light, 184, 410, 431, 444,

691 Abraham, M., 349 Absolute magnitude, 673 Absorptance, 409 Absorption, 418, 419, 670 Absorption coefficient, 363, 410, 671 Absorption lines, 676 Acceleration, 15, 42, 156, 220, 624 Acceptors, 610 Accuracy, 137 Ackermann, W. W., 740 Acoustic impedance, 359 Acoustical branches, 601 Acoustical circuits, 360 Acoustical constants, 357 Acoustics, 355 Action, 169, 209 Activity coefficients of aqueous electro

lytes, 648 Activity coefficients of gases, 645 Activity coefficients of nonelectrolytes,

646 Adams, D. P., 144 Adams, E. P., 106 Additional mass, 228 Adiabatic, 265, 281, 355 Adiabatic coefficient, 586 Adiabatic equilibrium, 678 Adiabatic lapse rate, 701 Adiabatic processes, 281 Adsorption, 655 Advection, 703 Aerodynamic center, 231 Aerodynamics, 218 Airfoil flow, 239 Airfoils, 232 Airship theory, 230 Aitken, A. C , 105 Algebra, 1 Algebraic equations, 3 Algebraic integrands, 22 Aller, Lawrence H., 668, 679

Almost free electrons, 624 Alternate hypothesis, 129 Alternating current, 329 Alternating tensor, 48 Ampere's law, 309 Amplitude and phase, 7 Analysis of variance, 134 Analytic function, definition of, 93 Analytic function, integrals of, 94 Analytic function, properties of, 94 Analytic functions, 93, 251 Anderson, 639 Anderson, R. L., 136 Angle between two lines, 7 Angle characteristic, 374 Angle of diffraction, 410 Angle of incidence, 366, 410 Angle of refraction, 366, 410 Angles, 5 Angular momentum, 156, 159 681 Angular velocity, 42 Anisotropy constants, 638 Anomalous diffusion, 721 Anomalous electron-moment correction,

152 Antiferromagnetics, 639 Antiferromagnetism, 640 Antisymmetric molecules, 467 Aperture defect, 445 Apparent additional mass, 229 Apparent magnitude, 673 Application to electron optics, 198 Approximation rules, 23 Arc length, 14 Arithmetic mean, 112 Arithmetic progression, 3 Associated Laguerre polynomials, 65,

512 Associated Legendre equation,, 100,

103 Associated Legendre's polynomials, 62,

247, 510 Associative matrix multiplication, 86 Astigmatic difference, 397 Astigmatism, 400

i

ii INDEX

Astronomical unit, 684 Astrophysics, 668 Asymmetric top molecules, 474, 497 Asymptotic expansion, 79 Atmospheric turbulence, 703 Atomic mass of deuterium, 148 Atomic mass of the electron, 150 Atomic mass of hydrogen, 148 Atomic mass of neutron, 148 Atomic mass of proton, 150 Atomic numbers, 527 Atomic oscillator, 182 Atomic specific heat constant, 151 Atomic spectra, 451, 668 Autobarotropic atmosphere, 701 Auxiliary constants, 148 Average, 82, 108, 112 Average deviation, 115 Avogadro's hypothesis, 296 Avogadro's number, 146, 149, 297,

425, 545, 623, 649 Axially symmetric fields, 437

BABCOCK, H. D . , 463 Babinet compensator, 428 Bacher, R. F., 463, 528, 543 Back, E., 459 Bacon, R. H., 695 Baker, James G., 365, 405, 679 Balmer series, 452 Bancroft, T. A., 136 Bardeen, J., 503, 631 Barrell, H., 463 Barrell and Sears formula, 462 Bartlett, M. S., 117 Bateman, H., 243 Bates, D . R., 463 Bauschinger, J., 695 Bauschinger-Stracke, 684 Bearden, J. A., 154 Beattie-Bridgeman, 270 Becker, R., 349 Becquerel formula, 430 Beers, N. R., 703 Beer's law, 419 Belenky, 554 Benedict, R. R., 354 Benedict, W. S., 503 Beranek, L. L., 364 Berek, M., 407 Bergeron, T., 704 Bergmann, L., 364 Bergmann, P. G., 169, 209

Bernoulli equation, 234 Bernoulli numbers, 69, 70, 603 Berry, F. A., 703 Berthelot, 270, 280 Bertram, S., 438, 449 Bessel equation, 101, 103 Bessel functions, 55, 63, 247, 528, 600 Bessel functions, asymptotic expressions,

58 Bessel functions, derivatives of, 59 Bessel functions, generating function

for, 59 Bessel functions, indefinite integrals in

volving, 59 Bessel functions, integral representation

of, 59 Bessel functions, modified, 59 Bessel functions of order half an odd

integer, 58 Bessel functions of order p, 56 Bessel functions of the first kind, 55 Bessel functions of the second kind, 56 Bessel functions of the third kind, 56 Bessel functions, recursion formula for,

59 Bessel's differential equation, 57 Beta decay, 541 Beta function, 68 Betatrons, 564, 578 Bethe, H. A., 528, 538, 543, 547, 561

632 Betti's formula, 718 Binding energy, 525, 526 Binomial distribution, 111, 125 Binomial theorem, 2 Biochemistry, 705 Biological phenomena, 705 Biological populations, 734 Biophysical theory, 722 Biophysics, 705 Biot Savart, 222, 321 Birge, R. T., 146, 154, 414, 463 Bivariate distribution, 116 Bjerknes, J., 704 Bjerknes, V., 704 Blackbody radiation laws, 418 Blair, 722 Blatt, J. M., 538, 543, 561 Bloch waves, 624 Bocher, M., 105 Bohm, D . , 524, 580 Bohr condition, 465 Bohr formula, 511

INDEX iii

Bohr frequency relation, 451 Bohr magneton, 151, 477, 623, 634 Bohr radius, 622 Bohr's frequency condition, 419 Bohr-Wilson-Sommerfeld, 517 Bollay, E., 703 Bollnow, O. F., 631 Boltzmann, L., 306 Boltzmann constant, 151, 209, 277,

292, 352, 425, 434, 468, 505, 602, 611, 631, 666, 668, 720

Boltzmann equation, 282, 294 Boltzmann factor, 634 Boltzmann formula, 668 Boltzmann's H theorem, 294 Borel's expansion, 79 Born, M., 349, 407, 424, 428, 430, 431,

631 Born approximation, 515, 547, 551 Bose-Einstein statistics, 284, 467 Bouguer's law, 419 Bound charge, 310 Boundary conditions, 102, 103, 409,

583, 714 Boundary layers, 242 Boundary potential, 721 Boundary value problems, 244 Boundary value problems, electrostatic,

314 Boundary value problems, magnetostatic,

325 Boutry, G . A . , 407 Bouwers, A . , 407 Bozman, W. R., 463 Bozorth, R. M., 640 Bradt, H. L., 561 Brand, L., 104 Brattain, W. H., 631 Breit, G . , 541 Brenke, W. C , 104 Brewster's angle, 367 Bridgman effect, 592 Bridgman, P. W., 264, 276 Brillouin function, 630, 634 Brillouin zone, 583, 624 Brouwer, D., 690, 695, 696 Brown, E. W., 693 Brown, O. E. and Morris, M., 104 Bruhat, G . , 407, 431 Brunauer-Emmett-Teller isotherm, 656 Brunt, D., 704 Buchdahl, H. A., 407 Buechner, W. W „ 580 Buoyancy, 219

Burger, H. C , 463 Burgers vector, 607 Burk, D. J., 741 Byers, H. R., 704

CADY , W. G . , 631 Calorimetric data, 642 Cameron, Joseph M., 107 Canonical coordinates, 519 Canonical equations, 161, 163, 189, 198 Canonical transformations, 164 Capacitance, 318 Capacitance, circular disk of radius, 319 Capacitance, concentric circular cylinders,

318 Capacitance, concentric spheres, 318 Capacitance, cylinder and an infinite

plane, 319 Capacitance, parallel circular cylinders,

318 Capacitance, parallel plates, 318 Capacitance, two circular cylinders of

equal radii, 319 Capacitors, 318, 328 Capacity, 619 Capillary waves, 235 Caratheodory, C , 407 Cardinal points, 385 Carnot engine, 265 Carnot cycle, 296 Carslaw, H. S., 105 Cartesian coordinates, 162, 165, 178,

211,245,484 Cartesian oval, 369 Cartesian surfaces, 368 Casimir, H. B. G . , 631 Catalan, M. A . , 460, 463 Catalysis, 661 Catalyst, 719 Catalyzed reactions, 708 Cauchy-Euler homogeneous linear equa

tion, 36 Cauchy formula, 392 Cauchy-Kowalewski, 191 Cauchy relations, 629 Cauchy-Riemann equations, 93, 251 Cauchy's integral formula, 94 Cauchy's mean value theorem, 15 Celestial mechanics, 680 Cell, 711 Cell polarity, 719 Center of gravity, 27 Center of pressure, 231 Central force, 157

iv INDEX

Central interactions, 531 Central limit theorem, 109 Chaffee, Emory L., 350 Chain rule, 11 Chandrasekhar, S., 289, 676, 679 Change of variables in multiple inte

grals, 26 Chaplygin, 239 Chapman, Sydney, 290, 306 Charge density, 190, 350 Charged particles, 537 Charge-to-mass ratio of the electron, 149 Charles' law, 296 Chemical composition, 678 Chemistry, physical, 641 Child's law, 352 Chi-square (x2) distribution, 110 Chretien, H., 407 Christoffel three-index symbol, 50, 211 Chromatic aberration, 388, 445 Churchill, R. V., 263 Circular accelerators, 564 Circular cylindrical coordinates, 348 Circular disk, 236 Circular membrane, 358 Circulation, 221, 231, 701 Circulation theorem, 701 Clairaut's form, 30 Clapeyron's equation, 269 Classical electron radius, 150 Classical mechanics, 155, 189 Clausius-Clapeyron equation, 650 Clausius equation, 377 Clemence, G. M., 686, 690, 695 Clock, 182 Cochran, W. G., 137 Cockcroft-Walton machines, 564, 571 Cohen, E. Richard, 145, 148, 154 Coleman, C. D., 463 Collar, A. R., 105 Collineation, 379 Collision cross-sections, 305 Collision energy, 298 Collision frequency, 298 Collision interval, 298 Collision problems, 512, 659 Collision theory, 659 Color excess, 674 Color index, 673 Coma, 399 Combination differences, 490 Combination principle, 451 Combination sums, 490 Combined operators, 47

Complementary function, 32 Complex exponentials, 21 Complex variable, 93, 227 Compliance coefficients, 584 Components, tensor, 46 Compound assemblies, 282 Compressibility, 587 Compressional modulus, 356 Compton effect, 187, 506, 552, 555 Compton wavelength, 150, 188 Comrie, L. J., 405 Concave grating, 415 Condon, E. U., 463, 679 Conducting sphere, 314 Conduction, 591 Conduction band, 609, 630 Conduction electrons, 640 Conductivity, 241, 308, 594, 613, 667 Conductivity tensor, 591 Conductors, 308 Confidence interval, 126, 128 Confidence limits, 108 Confluent hypergeometric function, 64 Conformal transformation, 228 Conjugate complex roots, 19 Conjugate tensor, 48 Conrady, A. E., 407 Consecutive reactions, 658 Conservation laws, 159, 196 Conservation of charge, 309 Conservation of energy, 157, 186 Conservation of mass, 698 Conservation of momentum, 186 Constants of atomic and nuclear physics,

145 Constraints, 157 Contact rectification, 617 Contact transformations, 167 Continuity equation, 714 Continuous absorption, 675, 676 Continuous systems, 174 Continuum, 673 Contravariant tensor, 179 Convection, 225 Convective potential, 195 Conversion factors of atomic and nuclear

physics, 145 Coolidge, A. S., 503 Cooling, 241 Corben, H. C , 177 Coriolis deflection, 700 Coriolis force, 697, 700 Correlation coefficient, 85, 133 Cosmic rays, 544

INDEX V

Cosmic rays, decay products, 559 Cotton-Mouton effect, 430 Couehe flow, 237 Coulomb energy, 625 Coulomb field, 524, 543 Coulomb potential, 511 Coulomb's law, 312 Coulomb wave function, 512 Couples, 681 Coupling, 455 Courant, R., 105, 243, 263 Cowling, T. G., 306 Cox, Arthur, 407 Cox, G. M., 137 Craig, C. C , 140 Craig, Richard A., 697 Cramer, H., 109 Crawford, B. L., 503 Crawford, F. H., 276 Critchfield, C. L. f 543 Critical energy, 549 Critical point, 270 Crossed electric and magnetic field, 441 Cross product, 40 Cross section, 513, 544, 545, 547, 630 Crystal mathematics, 581 Crystal optics, 427 Crystals, 417, 428 Cubic system, 585 Curie constant, 152, 633 Curie law, 633, 634 Curie temperature, 637 Curl, 43, 45, 220 Current, 350 Current density, 190, 193, 307, 311, 350 Current distributions, 321 Current intensity, 305 Current loops, 324 Curvature, 15, 42 Curvature of field, 400 Curvature tensor, 51 Curve fitting, 81 Curve of growth, 677 Curves and surfaces in space, 26 Curvilinear coordinates, 44, 246 Cyclic variables, 168 Cyclotron, 351, 564, 571, 578 Cyclotron frequency, 569 Cylindrical coordinates, 45, 165, 247 Cylindrical harmonics, 55 Cylindrical waves, 336 Czapski, S., 407

D'ALEMBERTIAN, 191 Dalton's law, 6 9 9 Damgaard, A . , 4 6 3 Damping constant, 3 5 7 Damping of oscillations, 5 7 1 Dayhoff, E . S. , 1 5 4 De Broglie wavelengths, 1 5 3 , 2 8 6 Debye-Huckel equation, 6 4 9 Debye length, 6 2 1 Debye temperature, 6 0 3 , 6 2 6 Debye-Waller temperature factor, 5 3 7 Decay constant, 5 2 9 Decibel, 3 5 6 Defay, R., 2 7 6 Definite integral, 2 2 Definite integrals of functions, 2 5 Deflection fields, 4 3 3 Deflection of light rays by the sun, 2 1 6 Degrees of freedom, 1 1 0 , 1 1 5 , 1 2 2 , 1 2 6 De Haan, B., 1 0 4 Demagnetization factor, 3 2 6 Dennison, D. M., 5 0 3 , 5 0 4 Density, 2 6 , 2 4 2 , 2 6 1 Density of states, 6 1 0 Derivatives, 1 0 Derived averages, 8 3 Derivatives of functions, 11 Determinants, 4 , 4 9 , 8 7 , 1 4 3 Deuteron, 5 3 3 Deviations, 8 3 , 3 7 1 Diagonal matrix, 8 9 Diamagnetism, 6 2 7 , 6 3 9 Diatomic molecule, 2 7 8 , 2 9 7 , 4 6 5 , 4 8 9 ,

6 6 9

Dickson, L. E . , 1 0 5 Dielectric constant, 1 9 3 , 3 0 8 , 4 1 0 , 4 2 2 ,

4 2 7 , 5 8 8 , 6 1 7 Dielectric media, 3 4 1 Dielectric sphere, 3 1 4 Dielectrics, 5 8 7 Dieterici, 2 7 0 , 2 8 0 Differential, 1 3 Differential calculus, 1 0 Differential equation and constant coeffi

cients, 3 3 Differential equations, 2 8 Differential equations and undetermined

coefficients, 34 Differential equations, classification of,

2 8 Differential equations, exact, 3 0 Differential equations, first-order and

first degree, 2 8

vi INDEX

Differential equations, first order and higher degree, 30

Differential equations, homogeneous, 30 Differential equations, inhomogeneous,

263 Differential equations, linear, 32 Differential equations linear in y, 29 Differential equations, numerical so

lutions, 39 Differential equations reducible to linear

form, 29 Differential equations, Runge-Kutta

method, 39 Differential equations, second order, 31 Differential equations, simultaneous, 36 Differential equations, solutions of, 28 Differential equations solvable for p, 30 Differential equations with variables sep

arable, 28 Differential equations, variation of pa

rameters, 35 Differentiation of integrals, 17 Diffraction, 416 Diffraction grating, 414 Diffraction of x rays, 417 Diffuse reflections, 366 Diffusion, 218, 225, 290, 300, 613, 663,

667, 707, 721 Diffusion coefficient, 614, 666 Diffusion flux, 707 Diffusion of vorticity, 225 Dilatation, 43 Dipole, 362, 425 Dirac, P. A. M., 524 Direct current, 328 Direction cosines, 6, 39 Dirichlet problem, 245 Discrimination, 726 Dislocation theory, 607 Dispersion, 424 Dispersion at a refraction, 371 Dispersion formulas, 391 Dispersion of air, 461 Dispersion of gases, 425 Dispersion of metals, 426 Dispersion of solids and liquids, 426 Dissipation, 224 Dissociating assemblies, 282 Dissociation energy, 481, 669 Dissociation equation, 669 Dissociation equation general, 283 Dissociation laws for new statistics, 287 Distortion, 400, 403 Distribution law, 85

Disturbed motion, 687 Disturbing function, 687 Ditchburn, R. W., 431 Divergence, 43, 220 Divergence theorem, 44 Donors, 610 Doppler broadening, 677 Doppler effect, 184, 431, 672 Dorgelo, H. B., 463 Dot product, 40 Double layer, 313 Double refraction, 409 Doublets, 456 Doublass, R. D., 144 Dow, W. G., 354 Drag, 232 Drift tubes, 571 Drude, Paul, 407, 431 Dual tensors, 206 Du Bridge, L. A., 354 Dugan, R. S., 679 Duhem-Margules, 275 Du Mond, Jesse W. M., 145, 154 Duncan, W. J., 105 Duncombe, R. L., 686, 695 Dunham, J. L., 503 Dunnington, F. G., 154 Durand, W. F., 243 Dwight, H. B., 104 Dyadics, 47 Dynamic equations, 224 Dynamical variable, 162 Dynamics, 186, 680 Dynamics of a free mass point, 188

EARTH , 690 Earth-moon system, 690 Earth's atmosphere, 697 Earth's rotation, 693 Ebersole, E. R., 740 Eccentric anomaly, 684 Echelon grating, 415 Eckart, C , 740 Ecliptic, 683 Eddington, A. S., 104, 209 Eddington approximation, 675 Eddy stresses, 698, 703 Eddy viscosity, 703 Edlen, B., 462 Ehrenfest's formula, 270 Ehrenfest's theorem, 507 Eigenfunctions, 261, 466, 470, 481, 487 Eigenvalues, 88, 89, 255

INDEX vii

Eikonal of Bruns, 372 Einstein, 178, 210, 212, 213 Einstein-Bose statistics, 520 Einstein effetcs, 214 Einstein's coefficients, 670 Eisenbud, L., 541 Eisenhart, C , 128 Eisenhart, L. P., 104 Elastic constant tensor, 629 Elastic constants, 584 Elastic moduli, 584 Elastic scattering, 539 Electrical conduction, 304, 626 Electrical constants, 422 Electric charge, 209, 307 Electric circuits, 328 Electric conductivity, 193, 613 Electric current, 307, 434 Electric density, 305 Electric-dipole transitions, 521 Electric displacement, 190, 308 Electric field, 190, 307, 350, 434, 438,

594 Electric multipoles, 312 Electric susceptibility, 308 Electrocaloric effect, 589 Electrochemical potential, 611 Electrodes, cylindrical, 352 Electrodes, infinite parallel planes, 352 Electrodynamics in moving, isotropic

ponderable media, 192 Electrodynamics of empty space, 208 Electrolytes, 308 Electromagnetic interactions, 544 Electromagnetic momentum, 331 Electromagnetic radiation, 330 Electromagnetic stress, 331 Electromagnetic theory, 307 Electromagnetic units, conversion tables,

346 Electromagnetic waves, 332 Electromagnetism, 307 Electromagnetism, the fundamental con

stants, 347 Electromotive force, 193 Electron accelerators, 563 Electron affinity, 353 Electron ballistics, 350 Electron guns, 433 Electronic charge, 146, 149, 545 Electronic energy, 279, 499 Electronic specific heat, 622 Electronic states, 499

Electronic transitions, 499 Electronics, 350 Electron microscope, 350, 433 Electron mirrors, 444 Electron optics, 198, 433 Electron orbit, 351 Electron pressure, 669 Electron rest mass, 149 Electrons, 547 Electron theory, 181, 621 Electro-optics, 428 Electrostatic generators, 564 Electrostatics, 311 Electrostatics of ionic lattices, 598 Elementary functions, 71 Elements of orbit, 683 Ellipse, 682 Elliptical coordinates, 247 Elliptically polarized wave, 411, 427 Elliptic motion, 684 Embryonic growth, 739 Emde, F., 106, 561 Emden's equation, 678 Emerson, W. B., 464 Emission, 418, 419 Emission of radiation, 670 Energy, 266, 281, 356 Energy density, 319 Energy density of sound, 356 Energy equation, 594 Energy generation, 678 Energy levels, 466, 469, 480, 489 Energy of electron, 351 Energy relations, 528 Energy states, 454 Enthalpy, 266, 647, 650 Entrance pupil, 400 Entropy, 209, 266, 277, 295, 587, 592,

595, 642, 648, 650, 667 Enumerative statistics, 125 Enzymatic reactions, 661 Enzyme, 715 Enzyme-substrate-inhibitor, 709 Eppenstein, O., 467 Epstein, D. W., 449 Epstein, P. S., 631 Equation of continuity, .221, 225, 226 Equation of state, 605, 698 Equation of state for an imperfect gas,

297 Equation of state for a perfect gas, 296 Equation of transfer, 675 Equation reducible to Bessel's, 57

viii INDEX

Equations of electrodynamics, 190 Equations of. a straight line, 7 , Equations of motion, 682 Equations of state, 270, 280 Equilibrium, 219, 641 Equilibrium constant, 641, 642 Equilibrium orbit, 569 Equilibrium radius, 569 Equipartition of energy, 296 Equipotential surfaces, 437 Equivalent conductivity, 664 Error distribution, 118, 119 Error integral, 69 Ertel, H., 704 Estimate, 108 Estimator, 108 Estimators of the limiting mean, 112 Ettingshausen coefficient, 593 Euclidean space, 211 Euler, 220 Euler angles, 91 Eulerian nutation, 693, 694 Euler-Lagrange equations, 163, 175, 176,

188, 197, 213 Euler-Maclaurin sum, 70 Euler's constant, 67 Euler's equation, 224 Event, 178 Exact differential, 13 Exchange energy, 626 Exchange operator, 526 Excitation potential, 668 Existential operator, 733 Exner, F. M., 704 Experiments, design of, 137 Exponential and hyperbolic functions,

integration of, 20 Exponential integral, 675 Exponential integrands, 21 Exponential law, 419 Exponentials and logarithms, 12 External fields, 476 External forces, 291 Ewald's method, 598

FABRY-PEROT INTERFEROMETER, 413 Faraday constant, 149, 643, 664 Faraday effect, 430 Faraday's law, 309, 329 F distribution, 111 Feenberg, E., 528 Fermat's principle, 163, 171, 100, 368 Fermat's principle fof electron optics, 436

Fermi coupling, 528 Fermi-Dirac statistics, 284, 467, 520,

610, 640 Fermi, E., 543 Fermi level, 612 Fermion, 531 Ferromagnetic materials, 309 Ferromagnetism, 636, 638 Field distribution, 437 Field of uniformly moving point charge,

194 Fieller, E. C , 140 Fine, H. B., 104 Fine-structure constant, 149, 454, 545 Fine-structure doublet separation in

hydrogen, 151 Fine-structure separation in deuterium,

151 Finney, D. J., 110 First Bohr radius, 150 First law of thermodynamics, 699, 706 First radiation constant, 151 Fisher, R. A., 123, 124, 131, 137 Fitting of straight lines, 116 Flatness of field, 402 Fletcher, H., 364 Flexible string, 357 Flow, 222, 227, 697 Fluctuations, 107, 354 Flux, 350, 409 Focal lengths, 381, 441 Focal points, 381 Focusing, 578 Foldy, Leslie L., 563, 580 Force, 223 Force on a charged particle, 563 Force on electrons, 350 Forsterling, K., 428, 431, 631 Forsyth, A. R., 104 Forsythe, G. E., 704 Fourier coefficients, 73, 625 Fourier integral, 260 Fourier integral theorem, 76 Fourier series, 54, 73, 257, 583 Fourier series, complex, 76 Fourier series, half-range, 74 Fourier series on an interval, 74 Fourier series, particular, 74 Fourier's theorem for periodic functions,

73 Fourier transforms, 77, 507 Fowler, R. H., 277, 289, 631 Franklin, Philip, 1, 77, 104, 105

INDEX IX

Frank, Nathaniel H., 307, 349, 426, 432 Frank, P., 105, 263 Fraunhofer diffraction, 416 Frazer, R. A., 105 Fredholm equation, 100 Fredholm integral equations, 96, 249 Free energy, 650 Free path, 298 Free surfaces, 233 Frenet formulas, 41 Frequency modulated cyclotron, 579 Fresnel biprism, 412 Fresnel-drag coefficient, 183,431 Fresnel equations, 342, 419 Fresnel formulas, 366 Fresnel integrals, 418 Fresnel mirrors, 412 Fresnel rhomb, 421 Fresnel zones, 418 Freundlich isotherm, 656 Freundlich-Sips isotherm, 656 Friction, 242 Fried, B., 464 Friedrichs, R. O., 243 Fringes, 413 Froude's rule, 235 Fry, D. W., 580 Fugacity, 274 Functions, miscellaneous, 66 Functions of sums and differences,

hyperbolic, 9 Functions of sums and differences of

angles, 5 Fundamental laws of geometrical optics,

370 Fundamental relativistic invariants, 209

GALVANOMAGNETIC EFFECT, 593 Gamma function, 56, 66, 67, 68 Gamma function, asymptotic expres

sions for, 67 Gamma function, logarithmic derivative

of, 67 Gamma function, special values of, 67 Gamma rays, 559 Gamow, G., 543 Gans, R., 449 Gardner, I. C , 407 Garfinkel, B., 696 Gas coefficients, 301 Gas constant, 292 Gas constant per mole, 148 Gaseous mixtures, • 273 Gas flow, 237, 239

Gas in equilibrium, 295 Gauge invariance, 191 Gaussian, 307 Gaussian units, 410 Gauss' law, 309 Gauss' method, 23 Gehrcke, E., 432 Geiger, H., 421, 422, 423, 432 General ellipsoid, 376 General relativity, 210 Geomagnetic effects, 559 Geometrical optics, 365, 409 Geometric progression, 3 Geophysics, 697 Geostrophic wind, 700 Gibbs adsorption equation, 656 Gibbs-Donnan membrane equilibrium,

653 Gibbs-Duhem equation, 647 Gibbs free energy, 707 Gibbs phase rule, 650 Gibbs thermodynamic potential, 266 Glaser, W., 436, 443, 444, 446, 449 Gleichen, A., 407 Goeppert-Mayer, M., 631 Goldberg, L., 464 Goldschmidt-Clermont, Y., 561 Goldstein, H., 177 Goldstein, S., 243, 704 Gooden, J. S., 580 Goranson, R. W., 272, 276 Gordy, W., 503 Goudsmit, S., 464 Gradient, 43, 220 Grashoff's rule, 236 Gravitational forces, 680 Gravitational potential, 210, 680 Gravity and capillarity, 235 Gravity waves, 235 Gray, A., 105 Gray, F., 438, 449 Green, J. B., 464 Green's function, 99, 102, 249, 311 Green's theorem in a plane, 44 Greisen, K., 554, 562 Greuling, E., 542 Griffith, B. A., 177 Group, 89 Group, Abelian, 90 Group, cyclic, 90 Group, isomorphic, 90 Group, normal divisor of, 90 Group representation, 91 Group theory, 89

X INDEX

Group theory, order, 90 Group theory, quotients, 90 Group, three-dimensional rotation, 91 Group velocity, 235 Gullstrand, A., 407 Guggenheim, E. A., 276, 289, 631 Gustin, W., 704 Guttentag, C , 740 Gwinn, W. D., 504 Gyromagnetic ratio, 152

HABELL, K. J., 407 Hagen-Rubens relation, 628 Hainer, R. M., 503 Haldane, J. B. S., 709, 740 Hall coefficient, 593, 614 Hall constant, 626 Hall effect, 614 Hall, H., 561 Halliday, D,, 580 Halpern, O., 561 Haltiner, G. J., 704 Hamel, G., 167 Hamiltonian, 162, 166, 506, 514 Hamiltonian function, 189 Hamiltonian operator, 530 Hamilton-Jacobi partial differential equa

tion, 168, 171 Hamilton's characteristic function, 372 Hamilton's principle, 163, 171 Hamilton, W. R , 407 Hankel functions, 56, 528 Hardy, A. C , 407 Harkins-Jura isotherm, 656 Harms, F., 422, 432 Harris, Daniel, 672 Harting's criterion, 392 Hartmann dispersion formula, 391 Hastay, M. W., 128 Haurwitz, B., 704 Heat, 209, 264 Heat absorbed, 266 Heat capacity, 241, 642 Heat conduction, 244, 261, 301 Heat energy, 292 Heat of vaporization, 667 Heavy-particle accelerators, 577 Heavy particles, 546, 547 Heisenberg, W., 163, 524, 555, 561 Heisenberg model, 637 Heisenberg uncertainty principle, 506 Heiskanen, W. A., 646 Heitler, W., 561 Hekker, F. 407 Helmholtz, 271

Helmholtz free energy, 266 Helmholtz-Lagrange formula, 382 Helmholtz's equation, 382 Helmholtz' theorem, 225 Henderson, R. S., 503 Henri, Victor, 708, 740 Henry's law, 275, 647 Herget, P., 683, 685, 695 Hermite equation, 101 Hermite functions, 65, 66, 481 Hermite polynomials, 65, 66, 509 Hermitian matrices, 87, 89, 92 Herrick, S., 695 Herring, Conyers, 581, 631 Herschel, Sir John, 688, 695 Hertz vector, 191, 332 Herzberg, G., 419, 432, 465, 503 Herzberg, L., 465 Herzberger, M., 394, 399, 407 Hess, S. L., 704 Heterogeneous isotropic media, 378 Heterogeneous systems, 275 Hewson, E. W., 704 Hexagonal system, 585 Hicks formula, 453 Hidden coordinate, 168 Hide, G. S. 580 High-energy phenomena, 544 Higher derivatives, 10 Hilbert, D., 105, 263 Hill, G. W., 694, 695 Hillier, J., 449 Hippisley, R. L., 106 Hittorf method, 664 Hjerting, 672 Hobson, E. W., 105 Hodograph, 222, 226 Hoffmann, 213 Hole, 609 Holmboe, J., 704 Homogeneous system, 5 Hopkins, H. H., 40$ l'Hospital's rule, 15 Hughes, A. L., 354 Humphreys, W. J., 704 Hund, F., 464 Hurwitz, A., 105 Hutter, R. G. E., 443, 449 Huygen's principle, 255, 340, 417 Hydrodynamic equation of motion, 697 Hydrodynamic equations, 707 Hydrodynamic "mobile operator," 299 Hydrodynamics, 218 Hydrogen fine structure, 200 Hydrogen ionization potential, 153

INDEX xi

Hydrostatic equation, 100 Hydrostatic equilibrium, 676, 678 Hydrostatic pressure, 294 Hydrostatics, 219 Hyperbola, 682 Hyperbolic functions, 8 Hyperbolic functions, derivatives of, 13 Hyperbolic motion, 686 Hyperfine structure, 460, 478 Hypergeometric and other functions, 62 Hypergeometric equation, 60, 102 Hypergeometric function, 60 Hypergeometric function, confluent, 64 Hypergeometric functions, contiguous, 61 Hypergeometric functions, elementary,

62 Hypergeometric functions, generalized,

63 Hypergeometric functions, special rela

tions, 62 Hypergeometric polynomials, 63 Hypergeometric series, 60 Hypothetical gases, 239

IMAGE CHARGES, 315 Image distance, 380 Image formation, 382 Image tubes, 433 Impact temperature increase, 242 Impedance, 328, 330, 354 Imperfect gas, 274, 280 Implicit function, 11 Improper integrals, 25 Impulsive acceleration, 572 Ince, E. L., 104 Inclination, 683 Incompressible flow, 225, 230 Increasing absolute value, 14 Indefinite integral, 17 Indefinite integrals of functions, 18 Indeterminate forms, 15 Index of refraction, 335, 365, 434 Index of refraction in electron optics, 436 Inductance, 326, 327 Induction acceleration, 571 Inertia, 236 Inertial forces, 158 Inertial system, 156 Inequalities, 24 Infeld, 213 Infinite products, 67, 73 Infrared, 493, 498 Inhibitors, 709

Injection, 578 Inner quantum number, 279 Integral calculus, 17 Integral equations, 96, 249 Integral equations with symmetric ker

nel, 97 Integral spectrum, 555 Integration by parts, 18 Intensities of spectral lines, 419 Intensity, 356, 409, 411 Interaction of populations, 733 Interfaces, 366 Interference, 409, 412 Interference of polarized light, 428 Intermediate coupling, 456 Internal electronic energy, 277 Internal reflection, 370 Interval estimation, 126 Interval estimators, 108 Inverse functions, 10, 11 Inverse hyperbolic functions, derivatives

of, 13 Inverse trigonometric functions, deri

vatives of, 12 Inversion spectrum, 495 Ion conductivity, 306, 666 Ionic strength, 649 Ionization loss, 546 Ionization potential, 668 Isentropic flow, 237 Isentropic potential, 224 Isobaric surface, 701 Isosteric surface, 701 Isothermal, 265 Isothermal coefficient, 586 Isothermal expansion, 240 Isotope effect, 482, 488 Isotopic spin, 531 Isotropic body, 585 Isotropic radiation, 670

JACOBIAN DETERMINANT, 26 Jacobi equation, 101 Jacobi identity, 163 Jacobi polynomials, 63 Jacobs, D. H., 408 Jahnke, E., 106, 561 Janossy, L., 561 Jeans, J. H., 306, 349 Jenkins, Francis A., 408, 409, 425, 432 Johnson, B. K., 408 Joule, 292 Joule-Thomson coefficient, 268

xii INDEX

KAPLON , M. F., 561 Karman's vortices, 234 Kayser, H., 451, 461 Kelvin equation, 654 Kendall, M. G., 126 Kennard, E. H., 306 Kepler ellipse, 215 Kepler's equation, 684 Kepler's first law, 682 Kepler's second law, 157, 683 Kepler's third law, 684, 690 Kernel, 97 Kerr constant, 430 Kerr electro-optic effect, 430 Kerst, D . W., 580 Kessler, K. G., 464 Kiess, C. C , 459, 464 Kinematics, 178, 220 Kinetic energy, 277 Kinetic stresses, 293 Kinetic theory of gases, 290 King, G. W., 503 Kingslake, R., 408 Kirchhoff, 417 KirchhofT's rules, 328 Kirchner, F., 154 Klapman, S. J., 562 Klein-Nishina formula, 552 Klein, O., 561 Knock-on probabilities, 545 Knopp, K., 105 Knudsen, V. O., 364 Koehler, J. S., 503, 632 Kohlschutter, A., 408 Konig, A., 408 Konopinski, E. J., 541 Koschmieder, H., 704 Kosters and Lampe, 462 Kosters, W., 464 Kostitzin, V. A., 741 Kron, G., 104 Kutta's condition, 231

L 2 , 531 Labs, D . , 679 Lagrange, 220 Lagrange equations, 161 Lagrange's law, 382 Lagrangian, 161, 199 Lagrangian function, 188 Lagrangian multiplier, 157 Laguerre equation, 101 Laguerre functions, 64, 65

Laguerre functions, orthogonality of, 65 Laguerre polynomials, 64, 512 Laguerre polynomials, generating func

tion of, 64 Laguerre polynomials, recursion for

mula for, 64 Lambda-type doubling, 501 Lambert-Beer law, 662 Lamb, H., 243, 704 Lamb, W., Jr., 154 Laminar flow, 703 Lampe, P., 464 Lanczos, C , 177 Landau formula, 640 Lande, A., 459 Lande factor, 429, 634 Lane, M. V., 209 Langevin formula, 633 Langevin function, 630, 636 Langevin-Pauli formula, 639 Langmuir, D . B., 436, 445, 449 Langmuir isotherm, 655 Laplace transforms, 77 Laplacian, 43 Laplace's equation, 51, 53, 93, 212, 225,

227, 228, 244 Laplace's integral, 53 Laporte, O., 201, 464 Latent heat, 595 Lattice, reciprocal, 583 Lattice types, 582 Laurent expansion, 94 Laurent expansion about infinity, 95 Laurent series, 95 Law of cosines, 8 Law of Helmholtz-Lagrange, 436 Laws of populations, 733 Laws of reflection, 369 Laws of refraction, 369 Law of sines, 8 Law of the mean, 15 Laws of thermodynamics, 265 Layton, T. W., 154 Learning, 729 Least action, 163 Least squares, 80, 147 Least-squares-adjusted values, 148, 149 Least-weights squares, 81 Legendre equation, 53 Legendre functions, asymptotic expres

sion, 53 Legendre functions, particular values

of, 52 Legendre polynomials, 51, 63, 527

INDEX xiii

Legendre polynomials, generating functions, 53

Legendre polynomials, orthogonality of, 53

Legendre's associated functions, 54 Legendre's associated functions, asymp

totic expression for w, 54 Legendre's associated functions, ortho

gonality, 55 Legendre's associated functions, partic

ular values of, 54 Legendre's associated functions, recur

sion formulas, 54 Legendre's differential equation, 51, 103 Legendre's functions, addition theorem,

55 Leibniz rule, 11 Length of arc, integration, 25 Lens equation, 436 Lenz, F., 443, 449 Leprince-Ringuet, L., 561 Lettau, H., 704 Levi-Civita, T., 214 Lienard-Wiechert potentials, 344 Lift coefficient, 231 Light, 335, 365 Light cone, 186 Light quanta, 412 Limiting mean, 113 Limit of error, 107 Line absorption, 676 Linear accelerators, 564, 579 Linear differential equation, 28 Linear equations, 4 Linear momentum, 159 Linear polyatomic molecules, 465, 490 Linear regression, 116, 120 Linear transformation, 143 Linear vector operator, 47 Linearity properties, 24 Linearly accelerated systems, 158 Line intensities, 456 Line of charge, 312 Line strengths, 672 Lineweaver, H., 741 Linfoot, E. H., 408 Liouville's theory, 166 Livingston, M. S., 538, 543, 580 Loeb, L. B., 306 Logarithmic differentiation, 12 Logarithmic-normal distribution, 110 Logarithms, 2 London equations, 593 London, F., 632

Longitudinal compressibility, 586 Longitudinal mass, 189, 351 Longley, R. W., 704 Lord Kelvin, 250 Lorentz, A. H., 209 Lorentz-covariant, 212 Lorentz-Fitzgerald contraction, 182 Lorentz-force, 197 Lorentz transformation, 179, 181, 185,

204 Loring, R. A., 464 Loschmidt's constant, 149 Loschmidt's number, 292, 297 Lotka, A., 741 Love, A. E. H., 632 Low-reflection coatings, 415 LS coupling, 455 Lummer-Gehrcke plate, 414 Lummer, O., 408, 432 Lunar parallax, 692 Lunisolar nutation, 694 Lunisolar precession, 694 Lyddane, R. H., 632 Lyman series, 452 Lynn, J. T., 464

M A C H , 238 Mach number, 236 Mach's rule, 236 Maclaurin's series, 16 Mac Robert, T. M., 105 Madelung, E., 106 Madelung's method, 600 Magnetic cooling, 629 Magnetic. dilution, 635 Magnetic displacement vector, 190 Magnetic dipole, 322 Magnetic field, 305, 438 Magnetic field intensity, 190, 308 Magnetic flux, 434 Magnetic guiding, 567 Magnetic induction, 307, 433, 594 Magnetic moment, 633, 636 Magnetic moment of the electron, 152 Magnetic monopole, 324 Magnetic multipoles, 323 Magnetic permeability, 193 Magnetic susceptibility, 308, 633 Magnetic vector potential, 161, 433, 437 Magnetism, 633 Magnetization, 308 Magnetocaloric effect, 629 Magnetomotive force, 193

xiv INDEX

Magneto-optics, 428 Magnetostatic energy density, 327 Magnetostatics, 320 Major axis, 683 Maloff, U., 449 Malone, T., 703 Malthusian law, 736 Malus, law of, 427 Malus, theorem of, 368 Mann, W. B., 580 Many-particle systems, 520 Marechal, A., 408 Margenau, H., 105 Martin, F. K,, 704 Mason, W . P . , 632 Mass, 26 Mass action, 641 Mass-energy conversion factors, 152 Massey, H. S. W., 562 Mass-luminosity law, 674, 679 Mass of planet, 692 Mathematical formulas, 1 Mathews, G. B., 105 Matrices, 85, 91 Matrices and linear equations, 87 Matrices column, 87 Matrices, diagonalization of, 88 Matrices in quantum mechanics, 518 Matrices, linear transformations of, 86 Matrices row, 87 Matrices, symmetry, 87 Matrices, unimodular, 91 Matrix, 85 Matrix addition, 86 Matrix multiplication, 86 Matrix notation, 46 Matrix products, 46 Matrix rank, 88 Matrix, transposed, 87 Matthiessen's rule, 591 Maxfield, F. A., 354 Maximum and minimum values, 14 Maxwellian tensor, 588 Maxwellian velocity distribution, 295, 615 Maxwell, J. G., 295, 306 Maxwell relations, 267 Maxwell stress tensor, 196 Maxwell's equations, 176, 191, 213, 282,

309, 320, 330, 409, 567, 594 Maxwell's law, 282 McConnell, A. J., 104 McKinley, W . A., 561 McLachlan, 105 Mean absolute error, 84

Mean anomaly, 683 Mean camber, 232 Mean elements, 690 Mean error, 107 Mean free path, 615 Mean free time, 615 Mean mass velocity, 291 Mean of range, 116 Mean orbit of the earth, 690 Mean orbit of the moon, 692 Mean shear modulus, 586 Mean-square deviation, 83 Mean values, 24 Measurement, 107 Measure of dispersion 84, 113 Measure of precision, 83, 137 Mechanical equivalent of heat, 292 Mechanical impedance, 357 Mechanics, 307 Mechanics, celestial, 680 Mechanics of a single mass point, 155 Media, 365 Median, 82, 108, 113 Meggers, W . F., 459, 462, 463, 464 Meggers' and Peters' formula, 461 Meissner, K. W., 414 Menzel, Donald H., 141, 277, 464, 679 Mercer kernel, 98 Meridional rays, 405 Merte, W., 390, 399, 403, 408 Meson decay, 558 Meson field theories, 532 Meson production, 557 Metabolism, 708, 711 Metabolites, 711, 719, 736 Metals, 308, 422, 423 Metal-semiconductor contact, 618 Metacenter, 219 Meteorology, 697 Method of images, 315 Metric tensor, 210 Meyer, C. F., 432 Michelson interferometer, 413 Michelson-Morley experiment, 182, 431 Microscope, 417 Microwave spectra, 503 Mie, G., 424 Mie-Gruneisen equation of state, 605 Milky Way, 674 Millman, J., 354 Minkowski's equation, 192 Minkowski "world," 178, 186 Mitchell and Zemansky, 672 Mixed characteristic, 374

INDEX XV

MKS, 307 Mobile charges, 611, 630 Mobility, 613 Model rules, 235 Modulus of viscosity, 241, 242 Molecular assemblies, 277 Molecular models, 301 Molecular spectra, 465 Molecular viscosity, 698, 703 Molecular weight, 292, 699, 706 Molecules with internal rotation, 497 Moliere, G., 556, 561 Molina, E. C , 112 Moment, 27 Momenta, 277 Moment of inertia, 27, 465, 469, 694 Moment of momentum, 222 • Momentum, 156, 218, 222, 544, 681 Monochromatic flux, 676 Mpnod, J., 737, 741 Montgomery, D. J. F., 562 Mood, A. M., 125 Moon, 692 Moore, Charlotte E., 451, 679 Morgan, J., 408 Morris, M. and Brown, O. E., 104 Morse function, 481 Morse, Philip M., 355, 364 Morton, G. A., 438, 440, 449 Motion of charged particle, 563 Mott, N. F., 562 Moulton, F. R., 683, 695 Muller-Pouillet, 432 Multinomial theorem, 2 Multiple angles, 6 Multiple arguments, 9 Multiple-hit processes, 659 Multiple regression, 124 Multiple roots, 3 Multiple scattering, 551 Multiplet splitting, 500 Multiplicity, 456 Multivariable systems, 271 Munk, Max M., 218 Munk's gas, 240 Murphy, G, M., 105 Mutual inductance, 326

NEEL , 639 Nernst's distribution law, 275 Nerve activity, 722 Nervous system, 725 Neumann problem, 245

Neumann series, 99 Neurone, 722, 725, 730 Neutrino, "528, 529 Neutron, 525, 534, 541 Neutron-proton scattering, 536 Neutron rest mass, 149 Newton, 3, 210, 646, 682 Newton's laws of motion, 155 Nicholson, J. W., 408, 432 Nielsen, H. H., 504 Nishina, Y., 561 Nodal points, 384 Nomogram, 141 Nomograph, 141 Noncentral interaction, 532 Non-gray material, 675 Nonlinear equations, 82 Nonrotational flow, 221 Nonuniform gas, 299 Nonviscuous fluids, 225 Nordheim, L. W., 562 Normal distribution, 83, 109, 115, 127 Normal equations, 81 Normalization, 509 Normal law, 83 Normal modes of a crystal, 601 Normal phase, 595 Normal probability functions, 109 Normal velocity surface, 427 Nuclear effects, 636 Nuclear interactions, 531, 541, 557 Nuclear magnetic moments, 460 Nuclear magneton, 152 Nuclear masses, 528 Nuclear phase shift, 527 Nuclear physics, 525 Nuclear radius, 525, 557 Nuclear spin, 478. Nuclear theory, 525, 528 Nuclear wave function, 528 Null hypothesis, 129 Nusselt number, 242

OBJECT DISTANCE, 380 Oblique refraction 395, 397 Oblique shock waves, 241 Observation equations, 81 Observations, 81 D'Ocagne, M., 144 Ohm's law, 193, 308, 591 Oliphant, M. L., 580 One-component systems, 266 Ongager's principle, 591

xvi INDEX

Optical branches, 601 Optical constants, 422, 423, 627 Optical depth, 675 Optical length, 367 Optical modes, 605 Optical path, 367 Optic axes, 427 Optics of moving bodies, 431 Orbital angular momentum, 524, 526, 530 Orbital electron capture, 529 Orbital quantum number, 279 Ordinary differential equation, 28 Orthogonal functions, 78 Orthogonal matrices, 87 Orthogonal polynomials, 123 Orthogonal tensor, 48 Orthonormal functions, 258 Oscillating dipole, 339 Oscillations, 570 Oscillator strength, 671 Osmotic pressure, 275, 653

PACKING-FRACTION, 525 Pair production, 552 Pannekoek, A . , 677 Parabola, 682 Parabolic coordinates, 45 Parabolic motion, 686 Paraboloid of revolution, 375 Paramagnetism, 633, 639 Paramagnetism, feeble, 639, 640 Paramagnetism, spin, 623 Paraxial path equations, 439 Paraxial ray equation, 439 Parsec, 673 Partial correlations, 124 Partial derivatives, 10 Partial differential equations, 28, 37 Partial pressures, 294 Partial regression coefficients, 124 Partially polarized light, 412 Particle acceleration, 571 Particle accelerators, 563 Partition functions, 277, 284, 668 Partition functions, convergence of, 284 Paschen, 452 Paschen-Back effect, 459 Path difference, 410 Path equation, 438 Pauli exclusion principle, 284 Pauling, F . , 524 Pauli spin matrices, 92, 523 Pauli's g-sum rule, 460

Pauli, W. Jr., 209, 464, 640 Peach, M., 632 Peltier effect, 592 Penetration depth, 594 Peoples, J. A . , 463 Perard, A . , 464 Perard's equation, 461 Perfect fluid, 219 Perfect gas, 219, 237, 269, 274, 280 Perihelion, 214 Period, 684 Permeability, 308, 410, 435, 720 Perrin, F . , 392, 407 Perturbations, 690 Petermann, A . , 154, 154a Peters, B., 561 Petzval surface, 402 Phase difference, 410 Phase motion, 576 Phase oscillations, 573, 574 Phase rule, 275, 650 Phase shifts, 534 Phase stability, 573 Phillips, H. B., 104 Photochemical equivalence, 661 Photochemistry, 661 Photoelectric disintegration, 533 Photoelectric emission, 353 Photomagnetic disintegration, 533 Photon, 412, 553 Photon emission, 540 Physical chemistry, 641 Physical optics, 409 Physiological stimulus, 725 Pi'cht, J., 439, 444, 449 Pickavance, T. G., 580 Pierce, B. O., 104 Pierpont, J., 105 Piezoelectric constants, 587 Piezoelectric crystals, 589 Piezoelectricity, 584, 587 Pitzer, K. S., 504 Planck function, 676 Planck relation, 187 Planck's constant, 149, 209, 278, 353,

451, 545, 610, 622, 631 Planck's law, 418, 670 Plane area, integration of, 25 Plane oblique triangle, 7 Plane-polarized light, 410, 420 Plane-polarized wave, 411 Plane right triangle, 7 Plane wave as a superposition of circular

waves, 260

INDEX xvii

Plane wave as superposition of spherical waves, 260

Plane wave, sound, 356 Poeverlein, H., 450 Point charge, 311 Points of inflection, 14 Poiseville flow, 236 Poisson brackets, 162, 168, 176 Poisson integral formula, 249 Poisson's distribution, 84, 112, 125 Poisson's equation, 212, 311, 693 Poisson's formula, 77

/Poisson's ratio, 608 Polar coordinates, 165 Polar crystals, 605 Polarization, 308, 421 Pole, 95 Polyatomic molecules, 483 Polynomials, derivatives of, 12 Polynomials, integration of, 18 Polytropic expansion, 226, 230, 238, 241 Ponderomotive equation, 197 Ponderomotive law, 213 Population, 736 Porter, J. G., 690, 695 Potential energy, 277, 486 Potential field, 157 Potential functions, 481 Potential scattering, 534 Potential theory, 245 Power, 330 Powers and quotients, 12 Power series, 70 Poynting vector, 196, 331, 411 Poynting's theorem, 330 p-n junction, 619 Prandtl-Meyer, 238 Prandtl number, 242 Precession, 694 Precision, 137 Pressure, 266, 355, 641, 643 Pressure gradient, 700 Pressure of a degenerate gas, 288 Preston, T., 408, 432 Price, D., 504 Prigogine, I., 276 Principal axes, 49 Principal planes, 383 Principal ray, 401 Probability, 83 Probable error, 84, 107, 115 Products by scalars, 40 Products, derivatives of, 11 Products, integration of, 21

Progressions, 3 Propagation of error, 138 Propeller disk, 233 Propeller efficiency, 233 Proportion, 2 Protein molecule, 715 Proton, 525 Proton moment, 152 Proton rest mass, 149 Pyroeiectricity,' 589 (

QUADRATIC EQUATIONS, 1 Quadrupole moment, 479 Quantum energy conversion factors, 152 Quantum mechanics, 505, 634 Quantum theory, 505, 634, 637 Quantum theory of dispersion, 426 Quarter-wave plate, 427

RACAH , G., 464 Radiant emittance, 409 Radiation, 361 Radiation, electromagnetic, 330 Radiation from a piston, 362 Radiation of electromagnetic waves, 339 Radiation length, 549 Radiation pressure, 670 Radiative equilibrium, 678 Radiative transfer, 675 Radicals, integration of, 20 Radius of curvature, 380, 563 Radius of electron orbit in normal H 1 ,

150 Radius of gyration, 27 Raman effect, 488,494 Raman spectrum, 468, 472, 476 Ramberg, Edward G., 433, 438, 440, 449 Randall, H. M., 504 Randall, R. H., 364 Random errors, 116 Random velocity, 292 Range, 546 Rank correlation coefficient, 126 Rankine flow, 227 Rankine-Hugoniot relations, 240 Raoult's law, 275 Rarita, W., 532 Rashevsky, N., 741 Rational functions, integration of, 18 Ray axes, 427 Rayleigh equation, 652 Rayleigh, J. W . S., 364 Rayleigh scattering, 424

xviii INDEX

Ray-tracing equations, 405 Ray velocity surface, 427 Reaction cross sections, 538 Reaction rate, 706 Reactions, 641 Reaction time, 726 Read, W. T., 632 Rebsch, R., 441, 445, 450 Reciprocal lattice, 583 Reciprocal potential, 227 Recknagel, A., 444, 450 Rectangular coordinates, 347 Rectangular distribution, 110 Red shift of spectral lines, 216 Reduced mass, 455, 466 Reflectance of dielectrics, 420 Reflectance of metals, 422 Reflection, 366, 371, 419, 606, 628 Reflection at moving mirror, 184 Reflection echelon, 415 Reflection of sound waves, 359 Reflexes, 729 Refraction, 366, 371, 381, 461 Refractive index, 183, 410, 427 Refractivity, 425 Reiner, John M., 705, 741 Rejection criterion, 130, 131 Relative motion, 42 Relativistic aberration, 446 Relativistic correction constant, 433 Relativistic degeneracy, 286 Relativistic electrodynamics, 189 Relativistic effects, 564 Relativistic force, 189 Relativistic invariants, 209 Relativistic mechanics, 186 Relativistic momenta, 197 Relativistic relations, 566 Relativistic wave equations, 522 Relativity, 182, 210 Relativity correction, 686 Relativity, field equations, 211 Relativity transformations, 558 Residual magnetization, 309 Residual rays, 606 Residues, 95 Resistance, 308 Resistivity, 308, 613 Resistors, 328 Resolving power, 414, 416 Resonance, 358, 570 Resonance reactions, 538 Rest energy, 544 Rest-mass, 187, 209, 351

Rest temperature, 209 Retarded potential, 263, 344 Reverberation time, 364 Reversible reactions, 659 Reynolds' number, 236, 242 Reynolds' rule, 236 Richards, J. A., 562 Richardson, M., 104 Richardson, R. S., 695 Richter, R., 390, 399, 408 Riemann, 69 Riemann-Christoffel curvature tensor, 51 Riemannian curvature tensor, 210, 211 Riemannian space, 50 Righi-Leduc coefficient, 593 Ring of charge, 312 Ritz combination principle, 452 Ritz formula, 453 Roberg, J., 562 Rodrigues' formula, 52 Rohr,M. von, 390, 399,408 Rojansky, V., 524 Rolle's theorem, 15 Rollett, J. S., 154 Room acoustics, 363 Root-mean-square, 83 Rose, M. E., 525 Rosenfeld, L., 543 Rosseland mean, 676 Rosseland, S., 679 Rossi, B., 408, 554, 562 Rotating coordinate systems, 159 Rotation, 43, 220, 465 Rotational eigenfunction, 499 Rotational energy, 277 Rotational quantum numbers, 469 Rotational structure, 502 Rotation and electron motion, 500 Rotation of the plane of polarization, 428 Rotation spectrum, 465, 468, 472 Rotation-vibration spectra, 489 Rotatory dispersion, 428 Ruark, A. E., 430, 432 Ruhemann, M. and B., 632 Rushbrook, G. S., 640 Russell, H. N., 464, 674, 679 Russell-Saunders coupling, 455 Rutherford formula, 514, 545 Rutherford scattering, 550 Rydberg and related derived constants,

153 Rydberg constant, 452 Rydberg equation, 451 Rydberg numbers, 148 Rydberg-Ritz formula, 453

INDEX xix

S\ 531 Sabine, 364 Sachs, R. G., 632 Sackur-Tetrode constant, 151 Saddle point, 80 Saha ionization formula, 283 Saturation corrections, 637 Saturation current, 353 Saver, R., 243 Scalar invariants, 49 Scalar potentials, 310, 322 Scalar product, 40 Scalars, 39 Scarborough, J. B., 696 Scattering, 424, 550 Scattering and refractive index, 425 Scattering by absorbing spheres, 425 Scattering by dielectric spheres, 424 Scattering cross sections, 538 Scattering length, 536 Scattering of a-particles, 537 Scattering of sound, 362, 363 Scattering, Rayleigh, 424 Scattering, Thomson, 424 Scheel, K., 421, 422, 423, 432 Scherzer, O., 450 Schiff, L. I., 505, 524 Schmidt series, 98 Schneider, W., 441, 445, 450 Schottky effect, 353, 618 Schottky, W., 276 Schrodinger constant, 150 Schrodinger, E., 173 Schrodinger wave equation, 506, 509,

512, 514, 624 Schuster, A., 408. 432 Schuster-Schwarzschild, 677 Schwarzschild, K., 214, 403, 408 Schwarzschild-Kohlschutter, 403 Schwarz's inequality, 24 Schwinger, J., 532, 562 Screening, 553 Sears, F. W., 408 Sears, J. E., 463 Secondary spectrum, 391 Second radiation constant, 151 Seebeck effect, 591 Seely, S., 354 Seidel aberrations, 399 Seidel, L. von, 408 Seitz, F., 632 Selection rules, 482, 487, 489, 491, 496,

497 Self-adjoint, 256

Self-exciting circuits, 728 Self-inductance, 326, 329 Sellmeier, 463 Semiconductors, 581, 609, 617 Semiconductors, «-type, 612 Semiconductors, />-type, 612 Sen, H. K., 679 Separated thin lenses, 387 Separation of variables, 246 Serber, R., 580 Series, 69 Series expansions in rational fractions, 72 Series, expansions of, 71 Series, integrals expressed in terms of, 72 Series formulas, 451 Shear modulus, 608 Shenstone, A. G., 454, 464 Shockley, W., 632 Shoock, C. A., 693 Shortley, G. H., 463, 464, 679 Shot noise, 354 Shower, 555 Shower maxima, 555 Shower theory, 553 Showers, spread of, 556 Siegbahn, 149 Simple oscillator, 357 Simple rational fractions, integration of,

18 Simpson's rule, 23 Simultaneity, 181 Simultaneous differential equations, 36 Sinclair, D., 424, 425 Sine and cosine of complex arguments, 10 Sine, cosine, and exponentials, 9, 10 Sink, 227, 712 Skew rays, 406 Skew-symmetric tensor, 48 Slater, J. C , 349, 426, 432, 580, 632 Slipstream velocity, 233 Slip vector, 607 Smart, W. M., 685, 696 Smith-Helmholtz equation, 382 Smith, J. H., 562 Smythe, W. R., 349 Snedecor, G. W. , I l l Snell's law, 341, 370 Snyder, H. S., 562 Solar parallax, 691 Solberg, H., 704 Solid state, 581 Solubility, 647 Sommerfeld, 454, 632 Sommerfeld's fine-structure constant, 146

X X INDEX

Sommerfeld's theory of the hydrogen fine structure, 200 •

Sommerfield, C M . , 154, 154a Sound, 355 Sound transmission through ducts, 359 Sources, 227, 712 Source strength, 361 Southall, J.P. C.., 408 Space absorption,, 673 Space charge, 352 Space-time continuum, 178, 210 Spangenberg, K. R., 354 Special accelerators, 578 Special theory of relativity, 178 Specific gravity, 220 Specific heat, 261, 281, 293, 297, 356,

595, 628 Specific heat conductivity, 261 Specific intensity, 670 Specific ionization, 547 Specific refractivity, 425 Spectra, 451, 489, 497 Spectroscopic stability, 636 Specular reflections, 366 Speed of light, 181, 209 Speiser, A., 105 Sphere, 236, 582 Sphere of charge, 312 Spherical aberration, 399 Spherical coordinates, 45, 247, 348 Spherical excess, 8 Spherical harmonics, 51 Spherical oblique triangle, 8 Spherical right triangle, 8 Spherical surface, 375 Spherical top molecules, 472, 496 Spherical wave as superposition of plane

waves, 260 Spherical waves, 337 Spin angular momentum, 520, 524 Spinning electron, 280 "Spin-only" formula, 635 Spin operator, 530 Spinor calculus, 201 Spin quantum number, 279 Spinors, mixed, 204 Sponer, H., 504 ^-scattering of neutrons, 535 Stability, 529, 570 Stagger ratio, 234 Standard deviation, 83, 85, 107, 113,

122, 127, 140 Standard distribution, 109 Standard volume of a perfect gas, 148

Stark broadening, 672 Stark effect, 430, 478 Stark energy, 635 Stars, 673 Stationary properties, 529 Statistical estimation, 108 Statistical mechanics, 264, 277, 290, 668 Statistical notation, 108 Statistical tests, 129 Statistical weight, 277, 419, 467, 4 71

631, 668 Statistics, 82, 107 Statistics, curve fitting, 124 Statistics of light quanta, 289 Statistics, the fitting of polynomials, 121 Staub, H., 562 Steady, normal shock, 240 Steady state, 295 Steepest descent, 80 Stefan-Boltzmann constant, 151, 676 Stefan-Boltzmann law, 419 Stefan's law, 271, 670 Stehle, P., 177 Steinheil-Voit, 408 Stellar equilibrium, 678 Stellar luminosity, 673 Stellar mass, 673 Stellar radius, 673 Stellar temperature, 673 Sterne, T. E., 696 Steward, G. C , 408 Stewart, J. Q., 679 Stirling's formula, 68 Stoichiometric combination, 709 Stokes' amplitude relations, 420 Stokes' law, 666 Stokes-Navier equation, 224 Stokes' theorem, 44 Stoner, E. C., 640 Stoner's model, 638 Stops, 400 Strain components, 584 Strain-tensor, 584 Stratton, J. A., 320, 349 Stream function, 221, 226 Streamlines, 251 Streamwise coordinate, 242 Streamwise distance, 242 Streamwise velocity, 242 Stress, 223, 293, 584 Stress distribution, 300 Stress on a conductor, 319 Stress tensor, 196, 293, 584, 608, 629,

706

INDEX xxi

Stromgren, B., 677 Strong, J., 408 Strongly ionized gas, 306 Strong magnetic lens, 442 Student's distribution, 111, 127 Sturm-Liouville differential equation,

100 Sturm-Liouville equation, 257 Subgroup, 90 Subsonic flow, 237 Suffix convention, 291 Summation convention, 46 Summational invariants, 294 Sum rule, 456 Sums of scalars, 40 Superconducting phase, 595 Superconductivity, 593 Superconductor, 594 Superconductors, multiply connected, 596 Superconductors, optical constants of,

597 Superconductors, resistance of, 597 Supersonic flow, 239 Surface current density, 311 Surface gravity, 674 Surface tension, 654, 715 Surface waves, 235 Susceptibility, 636 Susceptibility tensor, 588 Sutherland, 302 Sutton, O. G., 704 Symbolic vector, Del, 43 Symmetric molecules, 467 Symmetric top, 472 Symmetric top molecules, 492 Symmetry factor, 279 Symmetry properties, 470 Synaptic delay, 732 Synchrocyclotron, 564, 571, 579 Synchrotron, 564, 571, 579 Synge, J. L., 177, 372, 408

T A M M , 554 Taylor-Maccoll, 239 Taylor's series, 93, 138, 245, 734 Taylor's theorem, 16, 70, 313 Telescope, 416 Teller, E., 504, 632 Teller-Redlich rule, 488 Temperature, 266, 277, 292, 352, 434,

586, 631, 643, 667, 668, 699, 720 Temperature, stellar, 674 Tendency equation, 702

Tensor, fundamental, 50 Tensors, 46, 179 Tensors from vectors, 47 Tensors in w-dimensions, 49 Tensors of any rank, 50 Tensor, symmetric, 49 Tesseral harmonics, 53, 510 Test of homogeneity, 132 Test of linear law, 131 Test of normal distribution, 129, 130, 131 Theorem of Malus, 368 Theoretical fluid dynamics, 218 Theoretical intensities, 458 Thermal conduction, 290, 301, 626 Thermal conductivity, 302 Thermal diffusion, 304 Thermal noise, 354 Thermal release, 630 Thermal vibrations, 601 Thermionic emission, 353 Thermodynamic equation of state, 268 Thermodynamic equilibrium, 293 Thermodynamic functions, 602, 603 * Thermodynamics, 264, 277, 586, 642, 702 Thermoelectric effects, 614 Thermoelectric power, 591, 592, 614 Thermoelectricity, 591 Thermomagnetic effect, 593 Thermostatics, 264 Thick lens, 387 Thin lens, 386 Thirring, H., 632 Thompson cross section, 151 Thompson, H. D., 104 Thomson coefficient, 591 Thomson effect, 591 Thomson scattering, 424 Thomson, W., 250 Threshold energy, 538, 557 Throat flow, 239 Tidal waves, 235 Tietjens, O. G., 243 Time dilatation, 182 Tisserand, F., 694, 696 Tobocman, William, 307 Tolerance limits, 128 Torrey, H. C , 632 Torsion, 42, 586 Total cross section, 535 Total differential, 13 Total energy, 499 Total internal reflection, 421 Townes, C. H., 503 Track lengths. 554

xxii INDEX

Trajectories, 42 Transfer of radiation, 675 Transformation laws, 192 Transition probability, 419 Translation group, 582 Translations, 581 Transmittance, 410 Transmittance of dielectrics, 420 Transport phenomena, 662 Transverse mass, 189, 351 Trapezoidal rule, 23 Traps, 610 Traveling-wave acceleration, 572 Traveling-wave accelerator, 571 Triebwasser, S., 154 Trigonometric and hyperbolic functions,

9 Trigonometric functions, 5 Trigonometric functions, derivatives of,

12 Trigonometric functions, integration of,

19 Trigonometric integrands, 21 Trigonometric polynomials, 52 Trigonometry, 5 Triple scalar product, 41 True anomaly, 683 Trump, J. G., 580 Tschebycheff equation, 101 Tschebycheff polynomials, 63 Turnbull, H. W., 105 Two-dimensional flow, 231 Two-phase systems, 268 Two-photon decay, 559 Two-variable systems, 271

UHLENBECK, G. W., 201 Uncertainty principle, 506 Undisturbed velocity, 242 Uniform fields, 440 Uniformly convergent series, 98 Uniform motion, 186 Uniform rotation, 295 Unitary matrices, 87, 89, 91 Unitary tensor, 49 Unit cell, 582 Unit tensor, 48 Unpolarized light, 367 Unsold, A., 679 Urey, H. C., 430, 432

VACISEK, A., 415 Vacuum, 366

Valasek, J., 408, 417, 424, 425, 432 Valence band, 609 Vallarta, M. S., 562 Van Allen, J. A., 696 Vance, A. W., 449 Van de Graaff, 571, 580 Van der Waals, 270, 280, 297 Van't Hoff, 275, 653 Van Vleck, J. H., 426, 429, 432, 632,

633, 640 Vapor pressures, 273, 283, 654 Variance, 115 Variation of the elements, 688 Variation of the orbit, 689 Variational principles, 163, 212 Vector analysis, 39 Vector components, 39 Vector components, transformation of,

46 Vector derivatives, 41 Vector potentials, 310, 323 Vector product, 40 Vectors, 39 Velasco, R., 463 Velocity, 624 Velocity, absolute, 42 Velocity addition theorem, 183 Velocity distribution function, 192, 294 Velocity of light, 149, 365, 433, 545 Velocity of sound, 237 Velocity potential, 355 Velocity, relative, 42 Velocity vector, 15 Vening Meinesz, F. A„ 696 Verdet constant, 430 Vernal equinox, 683 Vibrating cylinder, 361 Vibration, 480 Vibrational eigenfunction, 499 Vibrational levels, 279 Vibrational structure, 502 Vibrations, 357 Vibration spectra, 480, 487 Virial equation of state, 270 Virtual temperature, 699 Viscosity, 219, 224, 232, 236, 290, 300,

302, 662, 667 Voigt, W „ 632 Voltage, 328 Volterra integral equations, 98 Volterra, V., 741 Volume, 266 Volumes, integration, 25 Von Mises, R., 105, 263

INDEX xxiii

Vortex motion, 234 Vorticity, 221, 225, 702

W A L D , A., 117 Walkinshaw, W., 580 Wsllis, W . A., 128 Water jumps and tidal waves, 235 Water vapor, 699 Watson, G. N., 105 Watts, H. M., 154 Wave equation, 244, 253, 255, 259, 260,

355, 409, 410 Wave functions, 285 Wave guides, 342 Wave impedance, 336 Wavelength, 409 Wave mechanics, 171 Wave number, 410 Wave optics, 365 Wave packet, 507 Waves, 235 Wave velocity, 410 Weak electron lenses, 441 Webster, A. G., 263 Weighted average, 113 Weight functions, 78 Weiss, 636 Weisskopf, V. F., 538, 543 Weiss molecular field, 636 Weizsacker, C. V., 528 Wendt, G., 445, 450 Wentzel-Kramers-Brillouin, 517 Weyl, H., 105, 209, 214 Wheeler, J. A., 541 White, H. E., 408, 425, 532, 464 Whitmer, C. A., 632 Whittaker, E. T., 105, 177, 399, 408 Wick, G. C , 562 Widder, D. V., 104 Wiedemann-Franz law, 626 Wien's displacement law, 151, 419, 670 Wien, W., 422, 432

Wigner, E. P., 105, 541 Williams, E. J., 551, 562 Williams, Robert W., 544 Wills, A. P., 104 Wilson, E. B., 524 Wilson, H. A., 214 Wilson, J. G., 562 Wilson, P. W., 740 Wing profile contours, 231 WKB, 517 Wood, P. J., 640 Wood, R. W., 432 Woolard, Edgar W., 680, 683, 696 Work, 266 World continuum, 210 World distance, 179 World-domain, 175 World interval, 179 World line element, 209 World point, 178 World vector, 193 Wronskian determinant, 32

YATES , F., 123, 131 Yost, F. L., 541 Young, G , 716, 741 Young's double slit, 412 Young's modulus, 586

ZATZKIS , Henry, 155, 178, 210, 244 Zeeman displacement per gauss, 151 Zeeman effect, 428, 429, 459, 476 Zeeman energy, 635 Zeeman patterns, 459 Zeeman separations, 634 Zemansky, M. W., 276 Zero-point energy, 480, 487 Zeta function, 69 Zinken-Sommer's condition, 401 Zonal harmonics, 51 Zworykin, V. K., 449

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Fundamental Formulas of Physics 1 (Menzel 0486605957)